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ELEMENTS 


ANALYTICAL    MECHANICS, 


W.  H.  C.  BARTLETT,  LL.D., 

PROFESSOR  OF    NATURAL  AND    EXPERLMENTAL    PHILOSOPHY    IN    THE    UNITED 
STATES  MILITARY  ACADEMY  AT  WEST  POINT, 

AND 

AUTHOR  OF  ELEMENTS  OF  MECHANICS,  ACOUSTICS  AND  OPTICS. 


NEW    YOKK: 
PUBLISHED    BY    A.    S.    BAFvNES    &    COMPANY 

No.    51    JOHN-STREET. 
CINCINNATI:    H.    W.    DERBY    A   CO. 

1S53. 


Entered  according  to  Act  of  Congress,  in  the  year  One  Tliousand 

Eight  Hundred  and  Fifty-three, 

By     W.    H.    C.    BART  LETT, 

In  the  Clerk's  Office  of  the  District  Court  of  the    United    States  for   the  Southern 

District  of  New-York. 


J.  P.  JONES  &  CO., 

STEREOTYPERS, 

1S3  William-st.,  New  York. 


"^ 


^Zlt^ 


PREFACE. 


V 


^ 


The  following  pages  were  mainly  prepared,  several  years  ago 
for  the  use  of  the  author's  class  in  the  United  States  Military 
Academy.  Their  publication  has  been  unavoidably  postponed 
to  the  present  time,  and  they  are  now  offered  to  the  public 
in  the  hope  that  they  may  contribute  something  to  lighten 
the  labor  which  every  student  must  encounter  at  the  threshold 
of  the  subject  of  which  it  is  their  purpose  to  treat. 

In  accordance  with  the  suggestions  of  much  experience  in 
the  business  of  teaching,  all  unnecessary  divisions  and  sub- 
divisions have  been  avoided.  Tlicy  too  often  divert  the  mind 
from  what  is  essential  to  that  which  is  merely  accidental,  and 
prevent  the  formation  of  those  habits  of  generalization  which 
alone  can  give  facility  in  acquiring  and  confidence  in  apply- 
in.*;-  any  branch  of  knowledge. 

^Mechanics  has  for  its  object  to  investigate  the  action  of- 
ferees upon  the  various  forms  of  bodies.  All  physical  phe- 
nomena are  but  the  necessary  results  of  a  perpetual  conflict 
of  equal  and  opposing  forces,  and  the  mathematical  formula 
expressive  of  the  laws  of  this  conflict  must  involve  the  M'hole 
doctrine  of  Mechanics.  The  study  of  Mechanics  should,  there- 
fore, be   made   to  consist   simply  in  the  discussion  of   tliis  for- 


IV 


PREFACE. 


mula,  and  in  it  sliould  be  sought  the  explanation  of  all  effects 
that  arise  from  the  action  of  forces. 

The  principle  of  classification  adopted,  is  that  suggested  by 
differences  in  the  physical  constitution  of  bodies,  and,  accord- 
ingly, the  subject  has  been  treated  under  the  heads  Mechanics 
OF  Solids  and  Mechanics  of  Fluids.  Much  time  and  space 
are  thus  saved,  the  attention  of  the  student  is  kept  con- 
stantly upon  his  subject,  and  the  discussion  divested  to  the 
utmost  of  all   specialties. 


CONTENTS. 


INTRODUCTION. 

PAGE. 

Preliminary  Definitions ^'■ 

Physics  of  Ponderable  Bodies ^^ 

Primary  Properties  of  Bodies 1^ 

Secondaiy  Properties 1" 

Force 20 

Physical  Constitution  of  Bodies 22 


PAKT    I. 

MECHANICS     OF    SOLIDS. 

Space,  Time,  Motion  and  Force 31 

Work ^^ 

Varied  Motion "*'- 

Equilibrium ^'^ 

The  Cord **'^ 

The  Muffle "^S 

Equilibrium  of  a  Rigid  System — Virtual  Velocities 50 

Principle  of  D'Alembert 55 

Free  Motion ^^ 

Composition  and  Resolution  of  Oblique  Forces 62 

Composition  and  Resolution  of  Parallel  Forces TS 


vi  CONTENTS, 

PAGE. 

Work  of  Resultant  and  of  Component  Forces ' 82 

Moments 84 

Resultant. 88 

Translation  of  General  Equations 91 

Centre  of  Gravity 93 

Centre  of  Gravity  of  Lines 97 

Centre  of  Gravity  of  Surfaces 102 

Centre  of  Gravity  of  Volumes 109 

Centrobaryc   Method H-l 

Centre  of  Inertia 1^^ 

Motion  of  the  Centre  of  Inertia 118 

Motion  of  Translation 1-0 

General  Theorem  of  Work  and  Living  Force 120 

Central  Forces 1-^ 

Stable  and  Unstable  Equilibrium 123 

Initial  Conditions,  Direct  and  Reverse  Problem 126 

Vertical  Motion  of  Heavy  Bodies 127 

Projectiles 1^^ 

Laws  of  Central  Forces 1^^ 

Rotary  Motion 1^^ 

Moment  of  Inertia,  Centre  and  Radius  of  Gyration 175 

Impulsive  Forces 1  ^^ 

Motion  under  the  Action  of  Impulsive  Forces 187 

Motion  of  the  Centre  of  Inertia 187 

Motion  about  the  Centre  of  Inertia 189 

Angular  Velocity 1^0 

Motion  of  a  System  of  Bodies 195 

Motion  of  Centre  of  Inertia  of  a  System 196 

Conservation  of  the  Motion  of  the  Centre  of  Inertia  of  a  System 197 

Conservation  of  Areas 1^^ 

Invariable  Plane -01 

Principle  of  Living  Force -O'" 

System  of  the  World 208 

Impact  of  Bodies -H 

Constrained  Motion  on  a  Surface -18 

Constrabed  Motion  on  a  Curve -"-0 

Constrained  Motion  about  a  Fixed  Point 246 

Constrained  Motion  about  a  Fixed  Axis 247 

Compound  Pendulum 

Motion  of  a  Body  about  an  Axis  under  the  Action  of  Impulsive  Forces 258 

Balistic  Pendulum , 259 


CONTEXTS.  vii 

PART    II. 

MECHANICS     OF    FLUIDS. 

PAGE. 

Introductory  Remarks 263 

Mariotte's  Law 265 

law  of  Pressure,  Density  and  Temperature 266 

Equal  Transmission  of  Pressure 268 

Motion  of  Fluid  Particles 270 

Equilibrium  of  Fluids 280 

Pressure  of  Hea\'y  Fluids 289 

Equilibrium  and  Stability  of  Floating  Bodies 295 

Specific  Gravity 304 

Atmospheric  Pressure 316 

Barometer 317 

Motion  of  Heavy  Incompressible  Fluids  in  Vessels 326 

Motion  of  Elastic  Fluids  in  Vessels 338 

PAET    III. 

APPLICATIONS     TO     SIMPLE    MACHINES,     PUMPS,    etc. 

General  Principles  of  all  Machines 345 

Friction 347 

Stifihess  of  Cordage 355 

Friction  on  Pivots 3G0 

Friction  on  Trunnions 365 

The  Cord  as  a  Simple  Machine ,  369 

The  Catenary 379 

Friction  between  Cords  and  Cjliudrical  Solids 381 

Inclined  Plane 383 

The  Lever 386 

Wheel  and  Axle 389 

Fixed  Pulley 391 

Movable  Pulley 394 

The  Wedge 400 

The  Screw 404 

Pumps 409 

The  Siphon 419 

The  Air-pump 421 


yiii  CONTENTS. 

I 

TABLES. 

PAGE. 

Table        I. — The  Tenacities  of  Different  Substances,  and  the  Resistances  which 

they  oppose  to  Direct  Compression 428 

Table  II. — Of  the  Densities  and  Volumes  of  "Water  at  Different  Degrees  of 
Heat,  (according  to  Stampfer),  for  every  2i  Degrees  of  Fahren- 
heit's Scale 430 

Table     IIL — Of  the  Specific  Gravities  of  some  of  the  most  Important  Bodies. .     431 

Table     IV. — Table  for  Finding  Altitudes  with  the  Barometer 434 

Table  V. — Co-efficient  Values,  for  the  Discharge  of  Fluids  through  thin 
Plates,  the  Orifices  being  Remote  from  the  Lateral  Faces  of  the 

Vessel 436 

Table      VI. — Experiments  on  Friction,  without  Unguents.     By  M.  Morin 431 

Table    VII. — Experiments  on  Friction  of  Unctuous  Surfaces.     By  M.  Morin 440 

Table  VIII. — Experiments  on  Friction  with  Unguents  interposed.     By  M.  Morin.     441 
Table     IX. — Of  Weights  necessary  to  Bend  different  Ropes  ai'ound  a  Wheel  one 

Foot  in  Diameter 443 

Table       X. — Friction  of  Trunnions  in  their  Boxes 445 


ELEMENTS 


ANALYTICAL  MECHANICS. 


INTEODUCTION. 

The  term  nature  is  employed  to  signify  the  assemblage  of  all 
the  bodies  of  tlie  universe ;  it  includes  whatever  exists  and 
is  the  subject  of  change.  Of  the  existence  of  bodies  we  are 
rendered  conscious  by  the  impressions  they  make  on  our  senses. 
Their  condition  is  subject  to  a  variety  of  changes,  whence  we 
infer  that  external  causes  are  in  operation  to  produce  them ;  and 
to  investigate  nature  with  reference  to  these  changes  and  their 
causes,  is  the  object  of  Pity  steal  Science. 

All  bodies  may  be  distributed  into  three  classes,  viz :  unorgan- 
ized or  inanimate^  organised  or  a/nimated^  and  the  heavenly  hodies 
or  primary  organisations. 

The  unorganized  or  inanimate  bodies,  as  minerals,  water,  air, 
form  the  lowest  class,  and  are,  so  to  speak,  the  substratum  for  the 
others.  These  bodies  are  acted  on  solely  by  causes  external  to 
themselves  ;  they  have  no  definite  or  periodical  duration  ;  nothing 
that  can  properly  be  termed  life. 

The  organised  or  animated  bodies,  are  more  or  less  perfect 
individuals,  possessing  organs  adapted  to  the  performance  of  cer- 
tain appropriate  functions.  In  consequence  of  an  innate  principle 


12  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

peculiar  to  them,  known  as  vitality,  bodies  of  this  class  are  con- 
stantly appropriating  to  themselves  unorganized  matter,  changing 
its  properties,  and  deriving,  by  means  of  this  process,  an  increase 
of  bulk.  They  also  possess  the  faculty  of  reproduction.  Tliey 
retain  only  for  a  limited  time  the  vital  principle,  and,  when  life 
is  extinct,  they  sink  into  the  class  of  inanimate  bodies.  The 
animal  and  vegetable  kingdoms  include  all  the  species  of  this 
class  on  our  earth. 

The  celestial  hodies,  as  the  fixed  stars,  the  sun,  the  comets, 
planets  and  their  secondaries,  are  the  gigantic  individuals  of  the 
universe,  endowed  with  an  organization  on  the  grandest  scale. 
Their  constituent  parts  may  be  compared  to  the  organs  possessed 
by  bodies  of  the  second  class ;  those  of  our  earth  are  its  conti- 
nents, its  ocean,  its  atmosphere,  which  are  constantly  exerting  a 
vigorous  action  on  each  other,  and  bringing  about  changes  the 
most  important. 

Tlie  earth  supports  and  nourishes  both  the  vegetable  and  animal 
world,  and  the  researches  of  Geology  have  demonstrated,  that 
there  was  once  a  time  when  neither  plants  nor  animals  existed  on 
its  surface,  and  that  prior  to  the  creation  of  either  of  these  orders, 
great  changes  must  have  taken  place  in  its  constitution.  As  the 
earth  existed  thus  anterior  to  the  organized  beings  upon  it,  we 
may  infer  that  the  other  heavenly  bodies,  in  like  manner,  were 
called  into  being  before  any  of  the  organized  bodies  which  pro- 
bably exist  upon  them.  Reasoning,  then,  by  analogy  from  our 
earth,  we  may  venture  to  regard  the  heavenly  bodies  as  the  pri- 
mary organized  forms,  on  whose  surface  both  animals  and  vege- 
tables find  a  place  and  support. 

Natural  PMlosojyhy,  or  Physics,  treats  of  the  general  proper- 
ties of  unorganized  bodies,  of  the  influences  which  act  upon  them, 
the  laws  they  obey,  and  of  the  external  changes  which  these 
bodies  undergo  without  afiecting  their  internal  constitution. 

Chemistry,  on  the  contrary,  treats  of  the  individual  properties 


INTRODUCTION.  13 

of  bodies,  hy  which,  as  regards  their  constitution,  they  may  be 
distinguished  one  from  another  ;  it  also  investigates  the  transfor- 
mations which  take  phicc  in  the  interior  of  a  body — transforma- 
tions by  which  the  substance  of  the  body  is  altered  and  remodeled ; 
and  lastly,  it  detects  and  classifies  the  laws  by  which  chemical 
changes  are  regulated. 

Natural  History,  is  that  branch  of  physical  science  which 
treats  of  organized  bodies ;  it  comprises  three  divisions,  the  one 
mechanical — the  anatomy  and  dissection  of  plants  and  animals  ; 
the  second,  chemical — animal  and  vegetable  chemistry  ;  and  the 
third,  explanatory — physiology. 

Astronomy  teaches  the  knowledge  of  the  celestial  bodies.  It  is 
divided  into  Sjyherical  and  Physical  astronomy.  The  former 
treats  of  the  appearances,  magnitudes,  distances,  arrangements, 
and  motions  of  the  heavenly  bodies ;  the  latter,  of  their  consti- 
tution and  physical  condition,  their  mutual  influences  and  actions 
on  each  other,  and  generally,  seeks  to  explain  the  causes  of  the 
celestial  phenomena. 

Again,  one  most  important  use  of  natural  science,  is  the  appli- 
cation of  its  laws  either  to  technical  purposes — mechanics,  tech- 
nical chemistry,  pharmacy,  c&c. ;  to  the  phenomena  of  the 
heavenly  bodies^pAys^'m^  astronomy;  or  to  the  various  objects 
which  present  themselves  to  our  notice  at  or  near  the  surface  of 
the  ^yiX^k—jyhysical  geograjphy,  meteorology — and  we  may  add 
geology  also,  a  science  which  has  for  its  object  to  unfold  the 
history  of  our  planet  fi'om  its  formation  to  the  present  time. 

Natural  philosophy  is  a  science  of  observation  and  expeAment, 
for  by  these  two  modes  we  deduce  the  varied  information  we 
have  acquired  about  bodies ;  by  the  former  we  notice  any 
changes  that  transpire  in  the  condition  or  relations  of  any  body 
as  they  spontaneously  arise  without  interference  on  our  part ; 
whereas,  in  the  performance  of  an   experiment,  we  purposely 


14:     ELEMENTS  OF  ANALYTICAL  MECHANICS. 

alter  the  natural  arrangement  of  things  to  bring  about  some  par- 
ticular condition  that  we  desire.  To  accomplish  this,  we  make 
use  of  appliances  called  philosophical  or  chemical  aj>]paratus,  the 
proper  use  and  application  of  which,  it  is  the  office  of  Exj^eri- 
'mental  Physics  to  teach. 

If  we  notice  that  in  winter  water  becomes  converted  into  ice, 
we  are  said  to  make  an  observation ;  if,  by  means  of  freezing 
mixtures  or  evaporation,  we  cause  water  to  freeze,  we  are  then 
said  to  perform  an  experiment. 

These  experiments  are  next  subjected  to  calculation,  by  which 
are  deduced  what  are  sometimes  called  -  the  laios  of  nature,  or  the 
rules  that  like  causes  will  invariably  produce  like  results.  To 
express  these  laws  with  the  greatest  possible  brevity,  mathematical 
symbols  are  used.  "When  it  is  not  practicable  to  represent  them 
with  mathematical  precision,  we  must  be  contented  with  infer- 
ences and  assumptions  based  on  analogies,  or  with  probable 
explanations  or  hypotheses. 

A  hypothesis  gains  in  probability  the  more  nearly  it  accords 
with  the  ordinary  course  of  nature,  the  more  numerous  the 
experiments  on  which  it  is  founded,  and  the  more  simple  the 
explanation  it  offers  of  the  phenomena  for  which  it  is  intended  to 
account. 

PHYSICS  OF  PONDERABLE  BODIES. 

§  1. — Tha  physical  properties  of  bodies  are  those  external  signs 
by  which  their  existence  is  made  evident  to  our  minds ;  the  senses 
constitute  the  medium  through  wliich  tliis  knowledge -is  com- 
municated. 

All  our  senses,  however,  are  not  equally  made  use  of  for  this 
purpose;  we  are  generally  guided  in  our  decisions  by  the  evidence 
of  sight  and  touch.  Still  sight  alone  is  frequently  incompetent, 
as  there  are  bodies  which  cannot  be  perceived  by  that  sense,  as, 
for  example,  all  colorless  gases  ;  again,  some  of  the  objects  of 
sight  are  not  substantial,  as,  the  shadow,  the  image  in  a  mirror. 


INTRODUCTIOX.  16 

spectra  formed  by  the  refraction  of  the  rays  of  light,  &c. 
Touch,  on  the  contrary,  decides  indubitably  as  to  the  existence 
of  any  body. 

The  properties  of  bodies  may  be  divided  inio primary  or  prin- 
cipal, and  secondary  or  accessory.  Tlie  former,  are  such  as  we 
find  common  to  all  bodies,  and  without  which  we  cannot  conceive 
of  their  existing ;  the  latter,  are  not  absolutely  necessary  to  our 
conception  of  a  body's  existence,  but  become  known  to  us  by 
investigation  and  experience. 

PKIMAEY  PEOPEKTIES. 

§  2. — The  primary  properties  of  all  bodies  are  extension  and 
impenetrability. 

Extension  is  that  property  in  consequence  of  which  every  body 
occupies  a  certain  limited  space.  It  is  the  condition  of  the 
mathematical  idea  of  a  body ;  by  it,  the  volume  or  size  of  the 
occupied  space,  as  well  as  its  boundary,  or  figure,  is  determined. 
The  extension  of  bodies  is  expressed  by  three  dimensions,  length, 
breadth,  and  thickness.  The  computations  from  these  data,  follow 
geometrical  rules. 

Impenetrahility  is  evinced  in  the  fact,  that  one  body  cannot 
enter  into  the  space  occupied  by  another,  without  previously 
thrusting  the  latter  from  its  place. 

A  body  then,  is  whatever  occupies  space,  and  possesses  exten- 
sion and  impenetrability.  One  might  be  led  to  imagine  that  the 
property  of  impenetrability  belonged  only  to  solids,  since  we  see 
them  penetrating  both  air  and  water  ;  but  on  closer  observation 
it  will  be  apparent  that  this  property  is  common  to  all  bodies  of 
whatever  nature.  If  a  hollow  cylinder  into  which  a  piston  fits 
accurately,  be  filled  with  water,  the  piston  cannot  be  thrust  into 
the  water,  thus  showing  it  to  be  impenetrable.  Invert  a  glass 
tumbler  in  any  liquid,  the  air,  unable  to  escape,  will  prevent  the 
liquid  from  occuj)ying  its  place,  thus  proving  the  impenetrability 


16 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


of   air.     The  diving-bell  affords  a  familiar  illustration  of   this 
property. 

The  difficulty  of  pouring  liquid  into  a  vessel  having  only  one 
small  hole,  arises  from  the  impenetrability  of  the  air,  as  the 
liquid  can  run  into  the  vessel  only  as  the  air  makes  its  escape. 
The  following  experiment  will  illustrate  this  fact : 

In  one  mouth  of  a  two- 
necked  bottle  insert  a  funnel 
«,  and  in  the  other  a  siphon  5 
the  longer  leg  of  which  is  im- 
mersed in  a  glass  of  water. 
Now  let  water  be  poured  into 
the  funnel  «,  and  it  will  be 
seen  that  in  proportion  as  this 
water  descends  into  the  vessel 
F^  the  air  makes  its  escape 
through  the  tube  5,  as  is 
proved  by  the  ascent  of  the 
bubbles  in  the  water  of  the 
tumbler. 


SECONDARY   PKOPERTIES. 

Tlie  secondary  properties  of  bodies  are  comp-essiUlity^  expansi- 
hility^  porosity,  divisibility,  and  elasticity. 

g  3. — Co7)vpressiUUty  is  that  property  of  bodies  by  virtue  of 
which  they  may  be  made  to  occupy  a  smaller  space :  and  expansi- 
bility is  that  in  consequence  of  which  they  may  be  made  to  fill  a 
larger,  without  in  either  case  altering  the  quantity  of  matter  they 
contain. 

Both  changes  are  produced  in  all  bodies,  as  we  shall  presently 
see,  by  change  of  temperature ;  many  bodies  may  also  be  reduced 
in  bulk  by  pressure,  percussion,  &c. 


INTRODUCTION.  17 

§  4. — Since  all  bodies  admit  of  compression  and  expansion,  it 
follows  of  necessity,  that  there  must  be  interstices  between  their 
minutest  particles ;  and  that  property  of  a  body  by  which  its 
constituent  elements  do  not  completely  fill  the  space  within  its 
exterior  boundary,  but  leaves  holes  or  pores  between  them,  is 
called  porosit I/.  The  pores  of  one  body  are  often  filled  with  some 
other  body,  and  the  pores  of  this  with  a  third,  as  in  the  case  of  a 
sponge  containing  water,  and  the  water,  in  its  turn,  containing 
air,  and  so  on  till  we  come  to  the  most  subtle  of  substances, 
etherj  which  is  supposed  to  pervade  all  bodies  and  all  space. 

In  many  cases  the  pores  are  visible  to  the  naked  eye  ;  in  others 
they  are  only  seen  by  the  aid  of  the  microscope,  and  when  so 
minute  as  to  elude  the  power  of  this  instrument,  their  existence 
may  be  inferred  from  experiment.  Sponge,  cork,  wood,  bread, 
&c.,  are  bodies  whose  pores  are  noticed  by  the  naked  eye.  The 
human  skin  appears  full  of  them,  when  viewed  with  the  magni- 
fying glass ;  the  porosity  of  water  is  shown  by  the  ascent  of  air 
bubbles  when  the  temperature  is. raised. 

§  5. — The  divisibility  of  bodies  is  that  property  in  consequence 
of  which,  by  various  mechanical  means,  such  as  beating,  pound- 
ing, grinding,  etc.,  we  can  reduce  them  to  particles  homogeneous 
to  each  other,  and  to  the  entire  mass ;  and  these  again  to  smaller, 
and  so  on. 

By  the  aid  of  matnematical  processes,  the  mind  may  be  led  to 
admit  the  infinite  divisibility  of  bodies,  though  their  practical 
division,  by  mechanical  means,  is  subject  to  limitation.  Many 
examples,  however,  prove  that  it  may  be  carried  to  an  incredible 
extent.  We  are  furnished  with  numerous  instances  among  nat- 
ural objects,  whose  existence  can  only  be  detected  by  means  of 
the  most  acute  senses,  assisted  by  the  most  powerful  artificial 
aids ;  the  size  of  such  objects  can  only  be  calculated  approxi- 
mately. 

Mechanical  subdivisions  for  purposes  connected  with  the  arts 

are  exemplified  in  the  grinding  of  corn,  the  pulverizing  of  sul- 

2 


18  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

pliiir,  charcoal,  and  saltpetre,  for  the  manufacture  of  gunpowder  ; 
and  Homeopathy  affords  a  remarkable  instance  of  the  extended 
application  of  this  property  of  bodies. 

Some  metals,  particularly  gold  and  silver,  are  susceptible  of  a 
very  great  divisibility.  In  the  common  gold  lace,  the  silver 
thread  of  which  it  is  composed  is  covered  with  gold  so  attenuated, 
that  the  quantity  contained  in  a  foot  of  the  thread  weighs  less 
than  go'oo  <^f  ^  grain.  An  inch  of  such  thread  will  therefore 
contain  j^ho  of  ^  g^'^i^  ^^  S^^^'^  ^^^  i^  ^^®  ^^^^  ^^  divided  into 
100  equal  parts,  each  of  which  would  be  distinctly  visible  to  the 
eye,  the  quantity  of  the  precious  metal  in  each  of  such  pieces 
would  be  yao^ooo  of  ^  grain.  One  of  these  particles  examined 
through  a  microscope  of  500  times  magnifying  power,  will  appear 
500  times  as  long,  and  the  gold  covering  it  will  be  visible,  having 
been  divided  into  3,600,000,000  parts,  each  of  which  exhibits  all 
the  characteristics  of  this  metal,  its  color,  density,  &c. 

Dyes  are  likewise  susceptible  of  an  incredible  divisibility. 
"With  1  grain  of  blue  carmine,  10  lbs.  of  water  may  be  tinged 
blue.  These  10  lbs.  of  water  contain  about  617,000  drops.  Sup- 
posing now,  that  100  particles  of  carmine  are  required  in  each 
drop  to  produce  a  uniform  tint,  it  follows  that  this  one  grain  of 
carmine  has  been  subdivided  62  millions  of  times. 

According  to  Biot,  the  thread  by  which  a  spider  lets  herself 
down  is  composed  of  more  than  5000  single  threads.  The  single 
threads  of  the  silkworm  are  also  of  an  extreme  fineness. 

Our  blood,  which  appears  like  a  uniform  red  mass,  consists  of 
small  red  globules  swimming  in  a  transparent  fluid  called  serum. 
The  diameter  of  one  of  these  globules  does  not  exceed  the  lOOOth 
part  of  an  inch  :  whence  it  follows  that  one  drop  of  blood,  such 
as  would  hang  from  the  point  of  a  needle,  contains  at  least  one 
million  of  these  globules. 

But  more  surprising  than  all,  is  the  microcosm  of  organized  nature 
in  the  Infusoria,  for  more  exact  acquaintance  with  which  we  are 
indebted  to  the  unwearied  researches  of  Ehrenberg.    Of  these  crea- 


INTRODUCTION.  19 

tures,  which  for  tho  most  part  we  can  see  only  by  the  aid  of  the 
microscope,  there  exist  many  species  so  small  that  millions  piled  on 
each  other  would  not  equal  a  single  grain  of  sand,  and  thousands 
might  swim  at  once  through  the  eye  of  the  finest  needle.  Tlie 
coats-of-mail  and  shells  of  these  animalcules  exist  in  such  prodi- 
gious quantities  on  our  earth  that,  according  to  Ehrenberg's  inves- 
tigations, pretty  extensive  strata  of  rocks,  as,  for  instance,  the 
smooth  slate  near  Bilin,  in  Bohemia,  consist  almost  entirely  of 
them.  By  microscopic  measurements  1  cubic  line  of  this  slate  con- 
tains about  23  millions,  and  1  cubic  inch  about  41,000  millions  of 
these  animals.  As  a  cubic  inch  of  this  slate  weighs  220  grains, 
187  millions  of  these  shells  must  go  to  a  grain,  each  of  which 
would  consequently  weigh  about  the  jji  millionth  part  of  a  grain. 
Conceive  further  that  each  of  these  animalcules,  as  microscopic 
investigations  have  proved,  has  his  limbs,  entrails,  <Scc.,  the  possi- 
bility vanishes  of  our  forming  the  most  remote  conception  of  the 
dimensions  of  these  organic  forms. 

In  cases  where  our  finest  insti'uments  are  unable  to  render  us 
the  least  aid  in  estimating  the  minuteness  of  bodies,  or  the 
degree  of  subdivision  attained;  in  other  words,  when  bodies 
evade  the  perception  of  our  sight  and  touch,  our  olfactory  nerves 
frequently  detect  the  presence  of  matter  in  the  atmosphere,  of 
which  no  chemical  analysis  could  afibrd  us  the  slightest  inti- 
mation. 

Thus,  for  instance,  a  single  grain  of  musk  diffuses  in  a  large 
and  airy  room  a  powerful  scent  that  frequently  lasts  for  yeai-s ; 
and  papers  laid  near  musk  will  make  a  voyage  to  the  East  Indies 
and  back  without  losing  the  smell.  Imagine  now,  how  many  par- 
ticles of  musk  must  radiate  from  such  a  body  every  second,  in 
order  to  render  the  scent  perceptible  in  all  directions,  and  you 
will  be  astonished  at  their  number  and  minuteness. 

In  like  manner  a  single  drop  of  oil  of  lavender,  evaporated  in  a 
spoon  over  a  spirit-lamp,  fills  a  large  room  with  its  fragrance  for 
a  length  of  time. 


20  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

§  (^.—Elasticity  is  the  name  given  to  that  property  of  bodies, 
by  virtue  of  which  they  resume  of  themselves  their  figure  and 
dimensions,  when  these  have  been  changed  or  altered  by  any 
extraneous  cause.  Different  bodies  possess  this  property  in  very 
different  degrees,  and  retain  it  with  very  unequal  tenacity. 

The  following  are  a  few  out  of  a  large  number  of  highly 
elastic  solid  bodies ;  viz.,  glass,  tempered  steel,  ivory,  whale- 
bone, &c. 

Let  an  ivory  ball  fall  on  a  marble  slab  smeared  with  some  col- 
orino-  matter.  The  point  struck  by  the  ball  shows  a  round  speck 
which  will  have  imprinted  itself  on  the  surface  of  the  ivory  with- 
out its  spherical  form  being  at  all  impaired. 

Fluids  under  peculiar  circumstances  exhibit  considerable  elas- 
ticity;  this  is  particularly  the  case  with  melted  metals,  more 
evidently  sometimes  than  in  their  solid  state.  The  following 
experiment  illustrates  this   fact   with  regard   to   antimony  and 

bismiith. 

Place  a  little  antimony  and  bismuth  on  a  piece  of  charcoal,  so 
that  the  mass  when  melted  shall  be  about  the  size  of  a  pepper- 
corn ;  raise  it  by  means  of  a  blowpipe  to  a  white  heat,  and  then 
turn  the  ball  on  a  sheet  of  paper  so  folded  as  to  have  a  raised 
edo-e  all  round.  As  soon  as  the  liquid  metal  falls,  it  divides  itself 
into  many  minute  globules,  which  hop  about  upon  the  paper  and 
continue  visible  for  some  time,  as  they  cool  but  slowly ;  the  points 
at  which  they  strike  the  paper,  and  their  course  upon  it,  will  be 
marked  by  black  dots  and  lines. 

The  recoil  of  cannon-balls  is  owing  to  the  elasticity  of  the  iron 
and  that  of  the  bodies  struck  by  them. 

FORCE. 

§  Y._"Whatever  tends  to  change  the  actual  state  of  a  body,  in 
respect  to  rest  or  motion,  is  called  a  force.  If  a  body,  for 
instance,  be  at  rest,  the  influence  which  changes  or  tends  to 
change  this  state  to  that  of  motion,  is  called  force.     Again,  if  a 


INTRODUCTION.  21 

"body  be  already  in  motion,  any  cause  which  urges  it  to  move 
faster  or  slower,  is  called  force. 

Of  the  actual  nature  of  forces  we  are  ignorant ;  we  know  of 
their  existence  only  by  the  effects  they  produce,  and  wdth  these 
we  become  acquainted  solely  through  the  medium  of  the  senses. 
Hence,  while  their  operations  are  going  on,  they  appear  to  us 
always  in  connection  with  some  body  which,  in  some  way  or 
other,  affects  our  senses. 

§  8. — We  shall  find,  though  not  always  upon  superficial  inspec- 
tion, that  the  approaching  and  receding  of  bodies  or  of  their  com- 
ponent parts,  when  this  takes  place  apparently  of  their  own 
accord,'  are  but  the  results  produced  by  the  various  forces  that 
come  under  our  notice.  In  other  words,  that  the  univei*sally  ope- 
rating forces  are  those  of  attraction  and  of  repulsion. 

§  9, — Experience  proves  that  these  universal  forces  are  at  work 
in  two  essentially  diflferent  modes.  Tliey  are  operating  either  in 
the  interior  of  a  body,  amidst  the  elements  which  compose  it,  or 
they  extend  their  influence  through  a  wide  range,  and  act  upon 
bodies  in  the  aggregate;  the  former  distinguished  as  Atomical 
or  Molecular  action^  the  latter  as  the  Attraction  of  gravitation. 

§  10. — Molecular  forces  and  the  force  of  gravitation,  often  co- 
exist, and  qualify  each  other's  action,  giving  rise  to  those  attrac- 
tions and  repulsions  of  bodies  exhibited  at  their  surfaces  when 
brought  into  sensible  contact.  This  resultant  action  is  called  the 
force  of  cohesion  or  of  dissolution,  according  as  it  tends  to  unite 
difi"erent  bodies,  or  the  elements  of  the  same  body,  more  closely, 
or  to  separate  them  more  widely. 

§  11. — Inertia  is  that  principle  by  which  a  body  resists  all 
change  of  its  condition,  in  respect  to  rest  or  motion.  If  a  body  be 
at  rest,  it  will,  in  the  act  of  yielding  its  condition  of  rest,  while 
under  the  action  of  any  force,  oppose  a  resistance ;  so  also,  if  a 
bodv  be  in  motion,  and  be  urged  to  move  faster  or  slower,  it  will, 


22  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

during  the  act  of  changing,  oppose  an  equal  resistance  for  every 
equal  amount  of  change.  We  derive  our  knowledge  of  this  prin- 
ciple solely  from  experience ;  it  is  found  to  be  common  to  all 
bodies ;  it  is  in  its  nature  conservative,  though  passive  in  charac- 
ter, being  only  exerted  to  preserve  the  state  of  rest  or  of  particu- 
lar motion  which  a  body  has,  by  resisting  all  variation  therein. 
Whenever  any  force  acts  upon  a  free  body,  the  inertia  of  the 
latter  reacts,  and  this  action  and  reaction  are  eqiial  and  contrary. 

%  12. — Molecular  action  chiefly  determines  the  forms  of  bodies. 
All  bodies  are  regarded  as  collections  or  aggregates  of  minute  ele- 
ipents,  called  ato7ns,  and  are  formed  by  the  attractive  and  repul- 
sive forces  acting  upon  them  at  immeasurably  small  distances. 

Several  hypotheses  have  been  proposed  to  explain  the  constitu- 
tion of  a  body,  and  the  mode  of  its  formation.  The  most  remark- 
able of  these  was  by  Boscovich,  about  the  middle  of  the  last  cen- 
tury. Its  great  fertility  in  the  explanations  it  afiords  of  the  prop- 
erties*of  what  is  called  tangible  matter,  and  its  harmony  with  the 
laws  of  motion,  entitle  it  to  a  much  larger  space  than  can  be 
found  for  it  in  a  work  like  this.  Enough  may  be  stated,  however, 
to  enable  the  attentive  reader  to  seize  its  leading  features,  and  to 
appreciate  its  competency  to  explain  the  phenomena  of  nature. 

1.  All  matter  consists  of  indivisible  and  inextended  atoms. 

2.  These  atoms  are  endowed  with  attractive  and  repulsive 
forces,  varying  both  in  intensity  and  direction  by  a  change  of  dis- 
tance, so  that  at  one  distance  two  atoms  attract  each  othei',  and  at 
another  distance  they  repel. 

3.  This  law  of  variation  is  the  same  in  all  atoms.  It  is,  there- 
fore, mutual ;  for  the  distance  of  atom  a  from  atom  J,  being  the 
same  as  that  of  h  from  a,  if  a  attract  5,  h  must  attract  a  with 
precisely  the  same  force. 

4.  At  all  considerable  or  sensible  distances,  these  mutual  forces 
are  attractive  and  sensibly  jproportional  to  the  square  of  the  dis- 
tance inversely.     It  is  the  attraction  called  gravitation. 

5.  In  the  small  and  insensible  distances  in  which  sensible  con- 


INTRODUCTION. 


23 


tact  is  observed,  and  whicli  do  Dot  exceed  the  lOOOtb  or  1500th 
part  of  an  incli,  there  are  many  alternations  of  attraction  and 
repulsion,  according  as  the  distance  of  the  atoms  is  changed. 
Consequently,  there  are  many  situations  within  this  narrow  limit, 
in  A\hich  two  atoms  neither  attract  nor  repel. 

6.  The  force  which  is  exerted  between  two  atoms  when  their 
distance  is  diminished  without  end,  and  is  just  vanishing,  is  an 
insuperable  repulsion,  so  that  no  force  whatever  can  press  two 
atoms  into  mathematical  contact. 

Such,  according  to  Boscovich,  is  the  constitution  of  a  material 
atom  and  the  whoU  of  its  constitution,  and  the  immediate  efficient 
cause  of  all  its  properties. 

Two  or  more  atoms  may  be  so  situated,  in  respect  to  position 
and  distance,  as  to  constitute  a  molecule.  Two  or  more  molecules 
may  constitute  ti  particle.     The  particles  constitute  a  hody. 

Now,  if  to  these  centres,  or  loci  of  the  qualities  of  what  is 
termed  matter,  we  attribute  the  property  called  inertia,  we  have 
all  the  conditions  requisite  to  explain,  or  arrange  in  the  order  of 
antecedent  and  consequent,  the  various  operations  of  the  physical 
world. 

Boscovich  represents  his  law  of  atomical  action  by  what  may 
be  called  an  exponential  curve.     Let  the  distance  of  two  atoms 


be  estimated  on  the  line  C  A  C,  A  being  the  situation  of  one  of 
them,  while  the  other  is  placed  anywhere  on  this  line.  When 
placed  at  «,  for  example,  we  may  suppose  that  it  is  attracted  by 
A,  with  a  certain  intensity.     We  can  represent  this  intensity  by 


24  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

the  length  of  tlie  line  i  Z,  perpendicular  to  A  C\  and  can  express 
the  direction  of  the  force,  namely,  from  i  to  A,  because  it  is 
attractive,  by  placing  i  I  above  the  axis  A  O.  Should  the  atom 
be  at  m,  and  be  repelled  by  A,  we  can  express  the  intensity  of 
repulsion  by  m  n,  and  its  direction  from  m  towards  G  by  placing 
■ni  n  below  the  axis. 

This  may  be.  supposed  for  every  point  on  the  axis,  and  a  curve 
drawn  through  the  extremities  of  all  the  perpendicular  ordinates. 
This  will  be  the  exponential  curve  or  scale  of  force. 

As  there  are  supposed  a  great  many  alternations  of  attractions 
and  repulsions,  the  curve  must  consist  of  many  branches  lying  on 
opposite  sides  of  the  axis,  and  must  therefore  cross  it  at  C*,  D', 
C'\  J)'\  &c.,  and  at  G.  All  these  are  supposed  to  be  contained 
within  a  very  small  fraction  of  an  inch. 

Beyond  this  distance,  whicti  terminates  at  G^  the  force  is 
always  attractive,  and  is  called  the  force  of  gravitation^  the  maxi- 
mum intensity  of  which  occurs  at  ^,  and  is  expressed  by  the 
length  of  the  ordinate  G'g.  Further  on,  the  ordinates  are  sensibly 
proportional  to  the  square  of  their  distances  from  J.,  inversely. 
The  branch  G'  G"  has  the  line  A  C\  therefore,  for  its  asymptote. 
Within  the  limit  A  O  there  is  repulsion,  which  becomes  infi- 
nite, when  the  distance  from  A  is  zero ;  whence  the  branch  O  i>» 
has  the  perpendicular  axis,  A  y,  for  its  asymptote. 

An  atom  being  placed  at  G^  and  then  disturbed  so  as  to  move 
it  in  the  direction  towards  A^  will  be  repelled,  the  ordinates  of  the 
curve  being  below  the  axis ;  if  disturbed  so  as  to  move  it  from 
J.,  it  will  be  attracted,  the  corresponding  ordinates  being  above 
the  axis.  The  point  G  is  therefore  a  position  in  which  the  atom 
is  neither  attracted  nor  repelled,  and  to  which  it  will  tend  to 
return  when  slightly  removed  in  either  direction,  and  is  called  the 
limit  of  gravitation. 

If  the  atom  be  at  6",  or  O'^^  (fee,  and  be  moved  ever  so  little 
towards  J.,  it  will  be  repelled,  and  when  the  disturbing  cause  is 
removed,  will  fly  back ;  if  moved  from  J.,  it  will  be  attracted 


INTRODUCTION. 


25 


a      A 


u       3 


and  return.  Hence  C\  0\  etc.,  are  positions  similar  to  C,  and  are 
called  limits  of  cohesion,  O  being  termed  the  last  limit  ofcoJie- 
sion.  An  atom  situated  at  any  one  of  these  points  will,  with  that 
at  A,  constitute  Vi.  perriianent  molecule  of  the  simj)lest  kind. 

On  the  contrary,  if  an  atom  be  placed  at  D',  or  D'\  tfcc,  and 
be  then  slightly  disturbed  in  the  direction  either  from  or  towards 
A,  the  action  of  the  atom  at  A  will  cause  it  to  recede  still  further 
from  its  first  position,  till  it  reaches  a  limit  of  cohesion.  The 
points  D',  D'\  (fee,  are  also  positions  of  indifierence,  in  which  the 
atom  will  be  neither  attracted  nor  repelled  by  that  at  A,  but  they 
differ  from  G,  G  C",  (fcc,  in  this,  that  an  atom  being  ever  so  little 
removed  from  one  of  them  has  no  disposition  to  return  to  it 
again  ;  these  points  are  called  limits  of  dissolution.  An  atom 
situated  in  one  of  them  cannot,  therefore,  constitute,  with  that  at 
A,  a  permanent  molecule,  but  the  slightest  disturbance  will  de- 
stroy it. 

It  is  easy  to  infer,  from  what  has  been  said,  how  tlu-ee,  four, 
etc.,  atoms  may  combine  to  form  molecules  of  different  orders  of 
complexity,  and  how  these  again  may  be  an-anged  so  as  by  their 
action  upon  each  other  to  form  particles.  Our  limits  will  not 
permit  us  to  dwell  upon  these  points,  but  we  cannot  dismiss  the 
subject  without  suggesting  one  of  its  most  interesting  conse- 
quences. 

According  to  the  highest  authority  on  the  subject,  the  sux  and 
other  heavenly  bodies  have  been  formed  by  the  gradual  subsi- 
dence of  a  vast  nehula  towards  its  centre.     Its  molecules  forced 


26  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

bj  their  gravitating  action  within  their  neutral  limits,  are  in  a 
state  of  tension,  which  is  the  more  intense  as  the  accumulation  is 
greater ;  and  the  molecular  agitations  in  the  sun  caused  by  the 
successive  depositions  at  its  surface,  make  this  bodj,  in  conse- 
quence of  its  vast  size,  the  principal  and  perpetual  fountain  of 
that  incessant  stream  of  ethereal  waves  which  are  now  generally 
believed  to  constitute  the  essence  of  light  and  heat.  The  same 
principle  furnishes  an  explanation  of  the  internal  heat  of  our 
earth  which,  together  with  all  the  heavenly  bodies,  would  doubt- 
less appear  self-luminous  were  the  acuteness  of  our  sense  of  sight 
increased  beyond  its  present  limit  in  the  same  proportion  that  the 
sun  exceeds  the  largest  of  these  bodies.  The  sun  far  transcends 
all  the  other  bodies  of  our  system  in  regard  to  heat  and  light,  and 
is  in  a  state  of  incandescence  simply  because  of  his  vastly  greater 
size. 

§  13. — The  molecular  forces  are  the  effective  causes  which 
hold  together  the  particles  of  bodies.  Through  them,  the 
molecules  approach  to  a  certain  distance  where  they  gain  a 
position  of  rest  with  respect  to  each  other.  The  power  with 
w^hich  the  particles  adhere  in  these  relative  positions,  is  called, 
as  we  have  seen,  cohesion.  Tliis  force  is  measured  by  the 
resistance  it  offers  to  mechanical  separation  of  the  parts  of 
bodies  from  each  other. 

The  different  states  of  matter  result  from  certain  definite 
relations  under  which  the  molecular  attractions  and  repulsions 
establish  their  equilibrium  ;  there  are  three  cases,  viz.,  two 
extremes  and  one  mean.  The  first  extreme  is  that  in  which 
attraction  predominates  among  the  atoms  ;  this  produces  the 
solid  state.  In  the  other  extreme  repulsion  prevails,  and  the 
gaseous  form  is  the  consequence.  Tlie  mean  obtains  when 
neither  of  these  forces  is  in  excess,  and  then  matter  presents 
itself  under  the  liquid  form. 

Let  A  represent  the  attraction   and  R  the   repulsion,   then 


INTRODUCTION.  27 

the  three  aggregate  forms  may  be  expressed  by  the  following 
form  ul  83 : 

A  >  Ji  solid, 

A  -C  li  gas, 

A  =  II   liquid. 

These  three  forms  or  conditions  of  matter  may,  for  the  most 
part,  be  readily  distinguished  by  certain  external  peculiarities; 
there  are,  however,  especially  between  solids  and  liquids,  so 
many  imperceptible  degrees  of  approximation,  that  it  is  some- 
times difficult  to  decide  where  the  one  form  ends  and  the 
other  begins.  It  is  further  an  ascertained  fact  that  many 
bodies,  (perhaps  all,)  as  for  instance,  water,  are  capable  of 
assuming  all  three  forms  of  aggregation. 

Thus,  supposing  th-at  the  relative  intensity  of  the  molecular 
forces  determines  these  three  forms  of  matter,  it  follows  from 
what  has  been  said  above,  that  this  term  may  vary  in  the 
same  body. 

Tlie  peculiar  properties  belonging  to  each  of  these  states 
will  be  explained  when  solid,  liquid,  and  aeriform  bodies  come 
severally  under  our  notice. 

§  14. — ^The  molecular  forces  may  so  act  upon  the  atoms  of 
dissimilar  bodies  as  to  cause  a  new  combination  or  union  of 
their  atoms.  This  may  also  produce  a  separation  between  the 
combined  atoms  or  molecules  in  such  manner  as  to  entirely 
change  the  individual  properties  of  the  bodies.  Such  efibrts 
of  the  molecular  forces  are  called  chemical  action ;  and  the 
disposition  to  exert  these  efforts,  on  account  of  the  peculiar 
state  of  aggregations  of  the  ultimate  atoms  of  different  bodies, 
chemical  affinity. 

§15. — Beyond  the  last  limit  of  gravitation,  atoms  attract 
each  other :  hence,  all  the  atoms  of  one  body  attract  each 
atom  of  another,  and  vice  versa:    thus   giving   rise    to    attrac- 


28     ELEMENTS  OF  ANALYTICAL  MECHANICS. 

tions  between  bodies  of  sensible  magnitudes  through  sensible 
distances.  The  intensities  of  these  attractions  are  proportional 
to  the  number  of  atoms  in  the  attracting  body  directly,  and 
to  the  square  of  the  distance  between  the  bodies  inversely. 

§16. — ^The  term  ^iniversal  gravitation  is  applied  to  this  force 
when  it  is  intended  to  express  the  action  of  the  heavenly 
bodies  on  each  other  ;  and  that  of  terrestrial  gravitation  or 
simply  gravity^  where  we  wish  to  express  the  action  of  the 
earth  upon  the  bodies  forming  with  itself  one  whole.  The 
force  is  always  of  the  same  kind  however,  and  varies  in 
intensity  only  by  reason  of  a  difference  in  the  number  of 
atoms  and  their  distances.  Its  effect  is  always  to  generate 
motion  when  the  bodies  are  free  to  move. 

Gravity,  then,  is  a  property  common  to  all  terrestrial  bodies, 
since  they  constantly  exhibit  a  tendency  to  approach  the 
earth  and  its  centre.  In  consequence  of  this  tendency,  all 
bodies,  unless  supported,  fall  to  the  surface  of  the  earth,  and 
if  prevented  by  any  other  bodies  from  doing  so,  they  exert  a 
pressure  on  these  latter. 

This  is  one  of  the  most  important  properties  of  terrestrial 
bodies,  and  the  cause  of  many  phenomena,  of  which  a  fuller 
explanation  will  be  given  hereafter. 

§17. — ^Tlie  mass  of  a  body  is  the  number  of  atoms  it  con- 
tains, as  compared  with  the  number  contained  in  a  unit  of 
volume  of  some  standard  substance  assumed  as  unity.  The 
unit  of  volume  is  usually  a  cubic  foot,  and  the  standard  sub- 
stance is  distilled  water  at  the  temperature  of  8 8°, To  Fahren- 
heit. Hence,  the  number  of  atoms  contained  in  a  cubic  foot 
of  distilled  water  at  38°,T5  Fahrenheit,  is  the  unit  of  mass. 

The  attraction  of  the  earth  upon  the  atoms  of  bodies  at  its 
surface,  imparts  to  these  bodies,  iceight ;    and  if  g   denote   the 


IXTRODUCTION.  29 

vreiglit  of  a  unit  of  mass,  J/,  the  number  of  units  of  mass  in 
the  entire  body,  tind  IF,  its  entire  weight,  then  will 

W  =  M  .g (1) 

§18. — Density  is  a  term  employed  to  denote  the  degree  of 
proximity  of  the  atoms  of  a  body.  Its  measure  is  the  ratio 
arising  from  dividing  the  number  of  atoms  the  body  contains, 
by  the  number  contained  in  an  equal  volume  of  some  standard 
substance  whose  density  is  assumed  as  unity.  The  standard 
substance  usually  taken,  is  distilled  water  at  the  temperature 
of  38°, 75  Fahrenheit.  Hence,  the  weights  of  equal  volumes  of 
two  bodies  being  proportional  to  the  number  of  atoms  they 
contain,  the  density  of  any  body,  as  that  of  a  piece  of  gold,  is 
found  by  dividing  its  weight  by  that  of  an  equal  volume  of 
distilled  water  at  38°,To  Fahrenheit. 

Denote  the  density  of  any  body  by  i>,  its  volume  by  F, 
and  its  mass  by  J/,  then  will 

M=  Y.D (ly 

which  in  Equation  (1),  gives 

TF  =  r.Z>.^ (2) 

§19. — ^That  branch  of  science  which  treats  of  the  action  of 
forces  on  bodies,  is  called  Mechanics.  And  for  reasons  which 
will  be  explained  in  the  proper  place,  this  subject  will  be 
treated  under  the  general  heads  of  Mechanics  of  Solids,  and 
Mechanics  of  Fluids. 


PART    I. 


MECHANICS    OF    SOLIDS 


SPACE,     TIME,    MOTION,    AND    FORCE. 

§  20. — Space  is  indefinite  extension,  without  limit,  and  contains  all 
bodies. 

§21. — Time  is  any  limited  portion  of  duration.  We  may  conceive 
of  a  time  which  is  longer  or  shorter  than  a  given  time.  Time  has, 
therefore,  magnitude,  as  well  as  lines,  areas,  &c. 

To  measure  a  given  time,  it  is  only  necessary  to  assume  a  certain 
interval  of  time  as  unity,  and  to  express,  by  a  number,  how  often 
this  unit  is  contained  in  the  given  time.  When  we  give  to  this 
number  the  particular  name  of  the  unit,  as  hour,  minute,  second,  &c., 
we  have  a  complete  expression  for  time. 

The  Instruments  usually  employed  in  measuring  time  are  clocks, 
chronometers,  and  common  watches,  which  are  too  well  known  to  need 
a  description  in  a  work  like  this. 

The  smallest  division  of  time  indicated  by  these  time-pieces  is  the 
second,  of  which  there  are  GO  in  a  minute,  3000  in  an  hour,  and 
86400  in  a  day  ;  and  chronometers,  which  are  nothing  more  than  a 
species  of  watch,  have  been  brought  to  such  perfection  as  not  to  vary 
in  their  rate  a  half  a  second  in   365  days,  or  31536000  seconds. 

Thus  the  number  of  hours,  minutes,  or  seconds,  between  any  two 
events    or   instants,    may    be    estimated   with    as    much    precision   and 


MECHANICS    OF    SOLIDS.  31 

ease  as  the  number  of  yards,  feet,  or  inches  between  the  extremities 
of  any  given   distance. 

Time  may  be  represented  by 
lines,  by  laying  off  upon  a 
civen  right  line  A  B,  the  equal 
distances  from  0  to  1,  1  to  2, 
2  to  3,  &c.,  each  one  of  these 
equal  distances  representing  the 
unit  of  time. 

A  second  is  usually  taken  as  the  unit  of  time,  and  a  foot  as  the 
linear  unit. 

S  22. — A  body  is  in  a  state  of  absolute  rest  when  it  continues  in  the 
same  place  in  space.  There  is  perhaps  no  body  absolutely  at  rest; 
our  earth  being  in  motion  about  the  sun,  nothing  connected  with  it; 
can  be  at  rest.  Rest  must,  therefore,  be  considered  but  as  a  relative 
term.  A  body  is  said  to  be  at  rest,  when  it  preserves  the  same 
position  in  respect  to  other  bodies  which  we  may  regard  as  fixed. 
A  body,  for  example,  which  continues  in  the  same  place  in  a  boat, 
is  said  to  be  at  rest  in  relation  to  the  boat,  although  the  boat  itself 
may  be  in  motion  in  relation  to  the  banks  of  a  river  on  whose  sur- 
face it  is  floating. 

§23. — A  body  is  in  motion  when  it  occupies  successively  different 
positions  in  space.  Motion,  like  rest,  is  but  relative.  A  body  is  in 
motion  when  it  changes  its  place  in  reference  to  those  which  we 
may  regard  as  fixed. 

Motion  is  essentially  continnons ;  that  is,  a  body  cannot  pass  from 
one  position  to  another  without  passing  through  a  series  of  interme- 
diate positions  ;  a  point,  in  motion,  therefore  describes  a  continuous 
line. 

When  we  speak  of  the  path  described  by  a  body,  we  are  to 
understand  that  of  a  certain  point  connected  with  the  body.  Thus, 
the  path  of  a  ball,  is  that  of  its  centre, 

g  24. — The  motion  of  a  body  is  said  to  be  curvilinear  or  rectilinear, 
according  as   the   path   described  is  a  curve  or  riff  lit  line.      Motion   is 


32  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

either  uniform  or  varied.  A  body  is  said  to  have  uniform  motion 
when  it  passes  over  equal  spaces  in  equal  successive  portions  of  time: 
and  it  is  said  to  have  varied  motion  when  it  passes  over  unequal 
spaces  in  equal  successive  portions  of  time.  The  motion  is  said  to 
be  accelerated  when  the  successive  increments  of  space  in  equal 
times  become  greater  and  greater.  It  is  relarded  when  these  incre- 
ments  become   smaller   and   smaller, 

§  25. —  Velocity  is  the  rate  of  a  body's  motion.  Velocity  is  mea- 
sured  by   the  length  of  path  described  uniformly  in  a  unit  of  time, 

§  26. — The  spaces  described  in  equal  successive  portions  of  time 
being  equal  in  uniform  motion,  it  is  plain  that  the  length  of  path 
described  in  any  time  will  be  equal  to  that  described  in  a  unit  of  time 
repeated  as  many  times  as  there  are  units  in  the  time.  Let  v  denote 
the  velocity,  t  the  time,  and  s  the  length  of  path  described,  then  will 

.         '  s  =  v.t, (3) 

If  the  position  of  the  body  be  referred  to  any  assumed  origin 
whose  distance  from  the  point  where  the  motion  begins,  estimated 
in  the  'direction    of  the   path   described,   be   denoted   by  S^  then  will 

s  ^  S  +  v.t (4) 

Equation  (3)  shows  that  in  unitbrm  motion,  the  space  described 
is  always  equal  to  the  ^:)rt>f/?/f^  of  the  time  into  the  velocity ;  that  the 
spaces  described  by  different  bodies  moving  with  different  velocities  during 
the  same  time,  are  to  each  other  as  the  velocities;  and  that  when  the 
velocities  are    the   same,    the  S2mces   are   to    each   other  as    the    times. 

§  27. — Differentiating   Equation    (3)    or    (4),    we   find 

ds  ... 

dt='-^ (^^ 

that   is   to    say,    the   velocity  is  equal   to  the  first  differential  co-efficient 
of  the    space    regarded  as  a  function   of  the  time. 

Dividing   both   members   of  Equation    (3)    by    t,   we  have 

f  = " («) 


MECHANICS    OF    SOLIDS.  33 

which  shows  that,  in  unifo^-m  motion,  the  veloci/i/  is  equal  to  (he  whole 
space   divided  by   the   time   in  tvhich   it   is   described. 

§  28. — Matter,    in    its    unorganized    state,  is   inaniinate  or  inert.     It 
cannot   give    itself   motion,   nor   can   it   change   of   itself   the    motion 
which  it  may  have  received. 
A  body  at  rest   will    forever 
remain    so    unless    disturbed 

by   something   extraneous    to  ? —^ ^ 

itself;    or  if  it  be  in  motion 
in  any   direction,   as   from    a 

to  6,  it  will  continue,  after  arriving  at  &,  to  move  towards  c  in  the 
prolongation  of  ab  ;  for  having  arrived  at  b,  there  is  no  reason  why 
it  should  deviate  to  one  side  more  than  another.  Moreover,  if  the 
body  have  a  certain  velocity  at  5,  it  will  retain  this  velocity  unaltered, 
since  no  reason  can  be  assigned  why  it  should  be  increased  rather 
than' diminished  in  the  absence  of  all  extraneous  causes. 

If  a  billiard-ball,  thrown  upon  the  table,  seem  to  diminish  its 
rate  of  motion  till  it  stops,  it  is  because  its  motion  is  resisted  by 
the  cloth  and  the  atmosphere.  If  a  body  throw^l  vertically  down- 
ward seem  to  increase  its  velocity,  it  is  because  its  weight  is  inces- 
santly urging  it  onward.  If  the  direction  of  the  motion  of  a  stone, 
thrown  into  the  air,  seem  continually  to  change,  it  is  because  the 
weight  of  the  stone  urges  it  incessantly  towards  the  surface  of  the 
earth.  Experience  proves  that  in  proportion  as  the  obstacles  to  a 
body's   motion  are  removed,  will  the  motion  itself  remain  unchanged. 

When  a  body  is  at  rest,  or  moving  with  uniform  motion,  its 
inertia   is   not   called  into    action. 

§  29. — A  force  has  been  defined  to  be  that  which  changes  or  tends 
to  change  the  state  of  a  body  in  respect  to  rest  or  motion.  Weight 
and  Heat  are  examples.  A  body  laid  upon  a  table,  or  suspended 
from  a  fixed  point  by  means  of  a  thread,  would  move  under  the 
action  of  its  weight,  if  the  resistance  of  the  table,  or  that  of  the 
fixed  point,  did  not  continually  destroy  the  efibrt  of  the  weight.  A 
body  exposed  to  any  source  of  heat  expands,  its  particles  recede 
from  each  other,  and  thus  the  state  of  the  body  is  changed. 

3 


34:  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

When  we  push  or  pull  a  body,  be  it  free  or  fixed,  we  experience 
a  sensation  denominated  pressure,  traction,  or,  in  general,  effort.  This 
effort  is  analogous  to  that  which  we  exert  in  raising  a  weight.  Forces 
are  real  pressures.  Pressure  may  be  strong  or  feeble;  it  therefore 
has  magnitude,  and  may  be  expressed  in  numbers  by  assuming  a 
certain  pressure  as  uyiity.  The  unit  of  pressure  will  be  taken  to  be 
that  exerted  by  the  weight  of  -g^,^  part  of  a  cubic  foot  of  distilled 
water,  at  38°,75,   and  is  called  a  pound. 

§30.— The  intensity  of  a  force  is  its  greater  or  less  capacity  to 
produce  pressure.  This  intensity  may  be  expressed  in  pounds,  or  in 
quantity  of  motion.  Its  value  in  pounds  is  called  its  statical  mea- 
sure •    in  quantity   of  motion,   its  dynamical   measure. 

§31. The  ^Jom<  of  apiplicaiion  of  a  force,  is  the    material  point  to 

which  the  force  may  be  regarded   as  directly  applied. 

§  32.— The  line  of  direction  of  a  force  is  the  right  line  which  the 
point  of  application  would  describe,   if  it  were  perfectly   free. 

§33. The  effect  of  a  force  depends  upon    its    intensity,  point   of 

application,  and  line  of  direction,  and  when  these  are  given  the  force 
is  known. 

R34. Two   forces   are    equal   when    substituted,  one    for    the  other, 

in  the  same  circumstances,  they  produce  the  same  effect,  or  when 
directly   opposed,   they  neutralize  each  other. 

§35. There    can   be   no    action   of  a   force   without  an    equal   and 

contrary  reaction.  This  is  a  law  of  nature,  and  our  knowledge  of  it 
comes  from  experience.  If  a  force  act  upon  a  body  retained  by  a 
fixed  obstacle,  the  latter  will  oppose  an  equal  and  contrary  resistance. 
If  it  act  upon  a  free  body,  the  latter  will  change  its  state,  and  in 
the  act  of  doing  so,  its  inertia  will  oppose  an  equal  and  contrary 
resistance.  Action  and  reaction  are  ever  equal,  contrary  and  simulta- 
neous. 

§  36. If  a  free  body  be  drawn  by  a  thread,  the  thread  will  stretch 

and  even  break  if  the  action  be  too  violent,  and  this  will  the  more 
probably  happen    in    proportion   as    the  body  is  more  massive.      If  a 


MECHANICS     OF    SOLIDS. 


35 


body   be   suspended   by   means   of  a   vertical   chain,    and   a   weighing 
spring  be   interposed   in   the   line  of  traction,  the   graduated  scale  of 
the  spring  will  indicate  the  weight  of  the 
body  when    the   latter   is   at  rest ;   but  if 
the    upper    end  of  the   chain   be    suddenly 
elevated,  the  spring  will  immediately  bend 
more    in    consequence    of    the    resistance 
opposed   by  the   inertia  of  the  body  while 
acquiring   motion.      When  the  motion    ac- 
quired becomes    uniform,  the    spring    will 
resume  and  preserve  the  degree  of  flexure 
which  it  had  at  rest.      If  now,  the  motion 
be  checl<ed  by  relaxing  the  effort   applied 
to  the  upper  end  of  the  chain,  the  spring 
will   unbend   and    indicate  a  pressure   less 
than   the   weight    of    the   body,   in    conse- 
quence of  the  inertia   acting    in   opposition    to   the   retardation.      The 
oscillations   of  the   spring  may  therefore  serve  to  indicate  the  varia- 
tions  in   the    motions   of    a  body,   and   the   energy   of   its   force   of 
inertia,   which   acts   against  or   with   a   force,    according    as   the  velo- 
city is  increased  or  diminished. 


§  37. — Forces  produce  various  effects  according  to  circumstances. 
They  sometimes  leave  a  body  at  rest,  by  balancing  one  another, 
through  its  intervention ;  sometimes  they  change  its  form  or  break 
it ;  sometimes  they  impress  upon  it  motion,  they  accelerate  or  retard 
that  which  it  has,  or  change  its  direction  ;  sometimes  these  effects  are 
produced  gradually,  sometimes  abruptly,  but  however  produced,  they 
require  some  definite  time,  and  are  effected  by  continuous  degrees.  If 
a  body  is  sometimes  seen  to  change  suddenly  its  state,  either  in 
respect  to  the  direction  or  the  rate  of  its  motion,  it  is  because  the 
force  is  so  great  as  to  produce  its  effect  in  a  time  so  short  as  to 
make  its  duration  imperceptible  to  our  senses,  yet  some  definite  por- 
tion of  time  is  necessary  for  the  change.  A  ball  fired  from  a  gun 
will  break  through  a  pane  of  glass,  a  piece  of  board,  or  a  sheet  of 
paper,  when  freely  suspended,  with  a  rapidity  so  great  as  to  call  into 


36 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


action  a  force  of  inertia  in  the  parts  which  remain,  greater  than 
the  molecular  forces  which  connect  the  latter    with  those  torn  away. 

In  such  cases  the  effects  are  obvious,  while  the  times  in  which 
they  are  accomplished  are  so  short  as  to  elude  the  senses  :  and  yet 
these  times  have  had  some  definite  duration,  since  the  changes,  corres- 
ponding to  these  effects,  have  passed  in  succession  through  their  differ- 
ent degrees  from  the  beginning  to  the  ending, 

§  38. — Forces  which  give  or  tend  to  give  motion  to  bodies,  are 
called  motive  forces.  The  agent,  by  means  of  which  the  force  is 
exerted,  is  called  a  Motor. 

§39. — The  statical  measure  of  forces 
may  be  obtained  by  an  instrument  called 
the  Dynamometer,  which  in  principle  does 
not  differ  from  the  spring  balance.  The 
dynamical  measure  will  be  explained  fur- 
ther  on. 

g  40. — When  a  force  acts  against  a  point 
in  the  surface  of  a  body,  it  exerts  a  pres- 
sure which  crowds  together  the  neighbor- 
ing particles  ;  the  body  yields,  is  compress- 
ed  and    its  surface  indented ;    the  crowded 

particles  make  an  effort,  by  their  molecular  forces,  to  regain  their 
primitive  places,  and  thus  transmit  this  crowdmg  action  even  to  the 
remotest  particles  of  the  body.  If  these  latter  particles  are  fixed  or 
prevented  by  obstacles  from  moving,  the  result  will  be  a  compression 
and  change  of  figure  throughout  the  body.  If,  on  the  contrary,  these 
extreme  particles  are  free,  they  will  advance,  and  motion  will  be  com- 
municated by  degrees  to  all  the  parts  of  the  body.  Tliis  internal  motion, 
the  result  of  a  series  of  compressions,  proves  that  a  certain  time  is 
necessary  for  a  force  to  produce  its  entire  effect,  and  the  error  of 
supposing  that  a  finite  velocity  may  be  generated  instantaneously. 
The  same  kind  of  action  will  take  place  when  the  force  is  employed 
to  destroy  the  motion  which  a  body  has  already  acquired;  it  will 
first  destroy  the  motion  of  the  molecules  at  and  nearest  the  point  of 
action,  and  then,  by  degrees,  that  of  those  which  are  more  remote 
in  the  order  of  distance. 


MECHAXICS     OF    SOLIDS.  37 

The  molecular  springs  cannot  be  compressed  without  reacting  in  a 
contrary  direction,  and  with  an  equal  effort.  The  agent  which  presses 
a  body  will  experience  an  equal  pressure;  reaction  is  equal  and  con- 
trary to  action.  In  pressing  the  finger  against  a  body,  in  pulling  it 
with  a  thread,  or  pushing  it  with  a  bar,  we  are  pressed,  drawn,  or 
pushed  in  a  contrary  direction,  and  with  an  equal  effort.     Two  weigh- 


ing springs  attached  to  the  extremities  of  a  chain  or  bar,  will  indicate 
the  same  degi'ee  of  tension  and  in  contrary  directions  when  made  to 
act  upon  each  other  through  its  intervention. 

In  every  case,  therefore,  the  action  of  a  force  is  transmitted  through 
a  body  to  the  ultimate  point  of  resistance,  by  a  series  of  equal  and 
contrary  actions  and  reactions  which  neutralize  each  other,  and  which 
the  molecular  sprmgs  of  all  bodies  exert  at  every  point  of  the  right 
line,  along  which  the  force  acts.  It  is  in  \-irtue  of  this  property  of 
bodies,  that  the  action  of  a  force  may  be  assumed  to  be  exerted  at 
any  2^oint  in  its  line  of  direction  within  the  boundanj  of  the  body. 

§41. — Bodies  being  more  or  less  extensible  and  compressible,  when 
interposed  between  the  force  and  resistance,  will  be  stretched  or 
compressed  to  a  certain  degree,  depending  upon  the  energy  with  which 
these  forces  act;  but  as  long  as  the  force  and  resistance  remain  the 
same,  the  body  having  attained  its  new  dimensions,  will  cease  to 
change.  On  this  account,  we  may,  in  the  investigations  which  follow, 
assume  that  the  bodies  employed  to  transmit  the  action  of  forces  from 
one  point  to  another,  are  inextensible  and  rigid. 

WORK. 

§42.— To  work  is  to  overcome  a  resistance  continually  recurring 
along  some  path.  Thus,  to  raise  a  body  through  a  vertical  height,  its 
weight  must   be  overcome   at   every  point  of  the  vertical  path.     If  a 


/^.    --  '■    "  ^ 


38  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

body  fall  through  a  vertical  height,  its  weight  overcomes  its  inertia  at 
every  point  of  the  descent.  To  take  a  shaving  from  a  board  with  a 
plane,  the  cohesion  of  the  wood  must  be  overcome  at  every  point 
along  the  entire  length  of  the  path  described  by  the  edge  of  the  chisel. 

§43. — The  resistance  may  be  constant,  or  it  may  be  variable.  In 
the  first  case,  the  quantity  of  tvork  performed  is  the  constant  resistance 
taken  as  many  times  as  there  are  points  at  which  it  has  acted,  and 
is  measured  by  the  product  of  the  resistance  into  the  path  described 
by  its  point  of  application,  estimated  in  the  direction  of  the  resistance. 
When  the  resistance  is  variable,  the  quantity  of  work  is  obtained  by 
estimating  the  elementary  quantities  of  work  and  taking  their  sum. 
By  the  elementary  quantity  of  work,  is  meant  the  intensity  of  the 
variable  resistance  taken  as  many  times  as  there  are  points  in  the 
indefinitely  small  path  over  which  the  resistance  may  be  regarded  as 
constant;  and  is  measured  by  the  intensity  of  the  resistance  into  the 
difierential  of  the  path,  estimated  in  the  direction  of  the   resistance. 

§  44. — In  general,  let  F  denote  any  variable  resistance,  arid  s  the 

path  described  by  its   point  of  application,  estimated  in  the  direction 

of  the  resistance;   then  will  the  quantity  of  work,  denoted  by   Q,  be 

given  by 

Q=fP.ds (7) 

which  integrated  between  certain  limits,  will  give  the  value  of  Q. 

§  45. — The  simplest  kind  of  work  is  that  performed  in  raising  a 
weight  through  a  vertical  height.  It  is  taken  as  a  standard  of  com- 
parison, and  suggests  at  once  an  idea  of  the  quantity  of  work 
expended   in    any  particular   case. 

Let   the  weight   be   denoted  by    W,  and  the  vertical  height  by  H ; 

then  will 

Q=  W.H (8). 

If  W  become  one  pound,  and  H  one  foot,  then  will 

e  =  1; 

and  the  unit  of  work   is,    therefore,  the   unit   of  force,    one   pound, 
exerted  over   the   unit  of  distance,  one  foot;    and   is  measured  by  a 


MECHANICS    OF    SOLIDS. 


39 


square  of  which  the  adjacent  sides  arc  respectively  one  foot  and  ono 
pound,  taken  from    the    same   scale  of  equal  parts. 

§46. — To  illustrate  the  use  of  Equation  (7),  let 
it  be  required  to  compute  the  quantity  of  work 
Decessary  to  compress  the  spiral  spring  of  the 
common  spring  balance  to  any  given  degree,  say 
from  the  length  AB  to  DB.  Let  the  resistance 
vary  directly  as  the  degree  of  compression,  and 
denote  the  distance  AB'  by  x ;    then  will 

F=  C.x; 

in  which  C  denotes  the  resistance  of  the  spring 
when  the  balance  is  compressed  through  the  dis- 
tance unity. 

This  value  of  F  in    Equation  (7),  gives 


Q 


fF.dx  =  fC.xdx  =  C.-+  C, 


which  integrated   between  the  limits   x  =  0    and   x  =  AD  =  a,  gives 


Q  =  C.—. 

Let   C=10  pounds,  a  =  S  feet;  then  will 

§  =  45     units  of  work, 

and  the  quantity  of  work  will  be  equal  to  that  required  to  raise 
45  pounds  through  a  vertical  height  of  one  foot,  or  one  pound 
through  a  height  of  45  feet,  or  9  pounds  through  5  feet,  or  5 
pounds  through  9  feet,  &c.,  all  of  which  amounts  to  the  same  thing. 

§47. — A  mean   resistance  is   that  which,  multiplied  into    the  entire 

path  described  in  the  direction  of  the  resistance,  Avill  give  the  entire 

quantity  of  work.  Denote  this  by  i?,  and  the  entire  path  by  s, 
and  from   the   definition,  we  have 


whence, 


R.s  =  fF.ds; 


(9). 


40 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


That  is,  the  mean  resistance  is  equal  to  the  entire  work,  divided 
by  the   entire  path. 

In  the  above  example  the  path  being  3  feet,  the  mean  resistance 
would   be   15   pounds. 

S48, — Equation  (7)  shows  that  the  quantity  of  work  is  equal  to 
the  area  included  between  the  path  s,  in  the  direction  of  the 
resistance,  the  curve  whose  ordinates  are  the  diiferent  values  of  F,  and 
the  ordinates  which  denote  the  extreme  resistances.  Whenever, 
therefore,  the  curve  which  connects  the  resistance  with  the  path  is 
known,  the  process  for  finding  the  quantity  of  work  is  one  of 
simple  integration. 

Sometimes   this   law   cannot    be   found,    and    the    intensity    of  the 
resistance   is   given   only  at  certain  points  of  the   path.     In  this  case 
we    proceed    as    follows,    viz.  :    At   the    several    points   of    the   path 
where   the   resistance   is    known,    erect    ordinates    equal   to   the   cor- 
responding  resistances,   and  connect    their    extremities   by   a    curved 
line ;   then  divide   the  path  described  into  any  eve?i  number  of  equal 
parts,  and  erect  the  ordinates 
at  the  points  of  division,  and 
at    the    extremities  ;    number 
the    ordinates    in    the    order 
of  the  natural  numbers;    add 
together    the  extreme   ordinates^ 
increase  this  sum  hy  four  times 
that  of  the  even  ordinates  and 
twice  that  of  the    uneven  ordi- 
nates, and  multiplt/  by  one-third 
of    the    distance    between    any 
two   consecutive   ordinates. 

Demonstration:  To  compute  the  area  comprised  by  a  curve,  any 
two  of  its  ordinates  and  the  axis  of  abscisses,  by  plane  geometry, 
divide  it  into  elementary  areas,  by  drawing  ordinates,  as  in  the 
last  figure,  and  regard  each  of  the  elementary  figures,  e-^  e^  r^  rj, 
«2  H  n  i''ii    <^c.,    as    trapezoids  ;    it    is    obvious    that    the    error   of 


MECHANICS     OF    SOLIDS. 


41 


r„nii2 


wi,  n,  4-  ^3  rs 


this  supposition  will  be  less,  in  proportion 
as  the  number  of  trapezoids  between  given 
limits  is  greater.  Take  the  first  two  trape- 
zoids of  the  preceding  figure,  and  divide  the 
distance  e,  e^  into  three  equal  parts,  and  at 
the  points  of  division,  erect  the  ordinates 
m  n,  nil  n^ ;  the  area  computed  from  the  three 
trapezoids  e^  m  n  rj,  m  m^  n,  n,  m,  e^  i\  n,,  will 
be  more  accurate  than  if  computed  from  the 
two  ej  e^  rj  r„   e.^  e-^  r^  r,. 

The   area  by  the   three   trapezoids  is 

e,r,  -\-  mn    ,              m  n  +  m^n^    , 
<?i  m  X  2 ^  ^^  "^' 2 ■  "^  "*'  ^' 

But  by   construction, 

e,  m  =  m  wii  =  ?«i  fg  =  ^  fi  ^3  =  I  ^1  fo, 
and  the   above   may   be   written, 

i  Ci  ^2  (fi  ri  +  2  w  w  +  2  m,  7?i  +  ^3  r^), 

but   in    the   trapezoid  w  Wj  n^  n, 

2  7nn  +  2  mi  ?;,  =  4  e^  r^,     very  nearly  ; 

whence   the  area  becomes 

i  f  1  <?o  (e,  ri  +  4  e.  r,,  +  ^3  rj)  ; 

the  area  of  the  next  two  tyipezoids  in  order,  of  the  preceding 
figure,  will   be 

i  ei  e,  (C3  r3  +  4  ^4  7-4  +  e,  r,)  ; 

and  similar  expressions  for  each  succeeding  pair  of  trapezoids. 
Taking  the  sum  of  these,  and  we  have  the  whole  area  bounded  by 
the    curve,  its   extreme   ordinates,  and    the   axis    of  abscisses ;    or, 

Q  =  ie,e,[e^r^  +  4e,r.,  +  2^3 7-3  +  Ac.r,  +  2e,r,  +  4e,r,  +  f,?"-!  •  (10), 
whence  the  rule. 


42  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

§49. — By  the  processes  now  explained,  it  is  easy  to  estinaate  the 
quantity  of  work  of  the  weights  of  bodies,  of  the  resistances  due  to 
the  forces  of  affinity  which  hold  their  elements  together,  of  their 
elasticity,  «Sic.  It  remains  to  consider  the  rules  by  which  the  quantity 
of  work  of  inertia  may  be  computed.  Inertia  is  exerted  only  during 
a  change  of  state  in  respect  to  motion  or  rest,  and  this  brings  us 
to   the  subject  of    varied   motion. 

VARIED    MOTION. 

§  50. — Varied  motion  has  been  defined  to  be  that  m  which  unequal 
spaces  are  described  in  equal  successive  portions  of  time.  In  this 
kind  of  motion  the  velocity  is  ever  varying.  It  is  measured  at  any 
given  instant  by  the  length  of  path  it  would  enable  a  body  to 
describe  in  the  first  subsequent  unit  of  time,  were  it  to  remain 
unchanged.  Denote  the  space  described  by  s,  and  the  time  of  its 
descriptions   by  t. 

However  variable  the  motion,  the  velocity  may  be  regarded  as 
constant  during  the  indefinitely  small  time,  dt  In  this  time  the 
body  will  describe  the  small  space  ds  ;  and  as  this  space  is  des- 
cribed uniformly,  the  space  described  in  the  unit  of  time  would, 
were  the  velocity  constant,  be  ds  repeated  as  many  times  as  the 
unit  of  time  contains  dt.  Hence,  denoting  the  value  of  the  velo- 
city at   any   instant  by   v,    we   have 

v  =  dsx^^', 

.  =  ^ (11). 

dt  ^     ^ 

§51. — Continual  variation  in  a  body's  velocity  can  only  be  pro- 
duced by  the  incessant  action  of  some  force.  The  body's  inertia 
opposes  an  equal  and  contrary  reaction.  This  reaction  is  directly 
proportional  to  the  mass  of  the  body  and  to  the  amount  of  change 
in  its  velocity ;  it  is,  therefore,  directly  proportional  to  the  product 
of  the  mass  into  the  increment  or  decrement  of  the  velocity.  The 
product  of  a   mass   into'  a   velocity,    represents  a  quantity  of  motion. 


MECHANICS    OF    SOLIDS.  43 

The  intensity  of  a  motive  force,  at  any  instant,  is  assumed  to  be 
measured  by 'the  quantity  of  motion  which  this  intensity  can  generate 
in   a  unit  of  time. 

The  mass  remaining  the  same,  the  velocities  generated  in  equal 
successive  portions  of  time,  by  a  constant  force,  must  be  equal  to 
each  other.  However  a  force  may  vary,  it  may  be  regarded  as 
constant  during  the  indefinitely  short  interval  dt\  in  this  time  it  will 
generate  a  velocity  dv^  and  were  it  to  remain  constant,  it  would 
generate  in  a  unit  of  time,  a  velocity  equal  to  dv  repeated  as  many 
times  as  dt  is  contained  in  this  unit;  that  is,  the  velocity  generated 
would   be   equal   to 

,      1        dv 
dv =  —  • 

dt        dt  ' 

and  denoting  the  intensity  of  the  force  by  P,  and  the  mass  by  J/, 
we  shall  have 

P  =  J/.  ^ (12). 

dt 

Again,  differentiating  Equation  (11),  regarding  t  as  the  independent 
variable,  we  get, 

d?s 
dv  =  -7-; 
dt  ' 

and   this,   in   Equation   (12),   gives 

P  =  J/.^ (13). 

dt-^ 

From  Equation  (11),  we  conclude  that  in  varied  motion,  the  velocity 
at  amj  instant  is  equal  to  the  Jirst  differential  co-efficient  of  the  space 
regarded  as  a  function  of  the  time. 

From  Equation  (12),  that  the  intensity  of  any  motive  force,  or  of 
the  inertia  it  develops,  at  any  instant,  is  measured  by  the  product  of 
the  mass  into  the  first  differential  co-efficient  of  the  velocity  regarded  as 
a  function  of  the  time. 

And  from  Equation  (13),  that  the  intensity  of  the  motive  force  or 
of  inertia,  is  measured  by  the  product  of  the  mass  into  the  second 
differential  co-efficient  of  the  space  regarded  as  a  function  of  the  lime. 


4A  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

§  52. — To  illustrate.     Let  there  be  the  relation 

s  =  aP-\-bP (14); 

required  the  space  described  in  three  seconds,  the  velocity  at  the  end 
of  the  third  second,  and  the  intensity  of  the  motive  force  at  the  same 
instant. 

Differentiating  Equation  (14)  twice,  dividing  each  result  by  dt,  and 
multiplying  the  last  by  31,  we  find 

d<! 

^  =  V  =  Safi  +  2ht     ■     '     .     .     (15), 
dt  ^     ' 

J/.^^  =  P  =  l/[Gai  +  26]  •     .     •     (16). 

Make  a  =  20  feet,  6  =z  10  feet,  and  t  =  3  seconds,  we  have, 
from  Equations  (14),  (15),  and   (16), 

s  =  20 .  33  +  10  .  32  z=  630   feet ; 

V  =  3  .  20  .  32  +  2 .  10 .  3  =  600   feet ; 

P  =  if  (6 .  20  . 3  +  2  .  10)  =:  380  .  M. 

That  is  to  say,  the  body  will  move«over  the  distance  630  feet  in 
three  seconds,  will  have  a  velocity  of  600  feet  at  the  end  of  the 
third  second,  and  the  force  will  have  at  that  instant  an  intensity 
capable  of  generating  in  the  mass  J/,  a  velocity  of  380  feet  in  one 
second,  were  it  to  retain  that  intensity  unchanged. 

§  53. — Dividing   Equations   (12)    and,  (13)   by   M,  they  give 

(17), 


p 

M~ 

dv 
dt 

P 
M~ 

dh 
dfi 

(18). 


The  first  member  is  the  same  in  both,  and  it  is  obviously  that 
portion  of  the  force's  intensity  which  is  impressed  upon  the  unit  of 
mass.  The  second  member  in  each  is  the  velocity  impressed  in  the 
unit  of  time,  and  is  called  the  acceleratioyi  due  to  the  motive  force. 


MECHANICS    OF    SOLIDS.  45 

§54, — From   Equation   (11)  we   have, 

ds  =  v.di (19); 

multiplying  this  and  Equation  (12)  together,  there  Avill  result, 
F.ds  =  M.v.dv       .     .     .     .     (20), 


and   integratinjj, 


fF.ds=-i- (21). 


The  first  member  is  the  quantity  of  work  of  the  motive  force, 
which  is  equal  to  that  of  inertia ;  the  product  3f.  v^,  is  called  the 
living  force  of  the  body  whose  mass  is  M.  "Whence,  we  see  that 
the  ivork  of  inertia  is  equal  to  half  the  living  force  ;  and  the  living 
force  of  a  body  is  double  the  quantitrj  of  work  cxjjcnded  by  its  inertia 
while  it  is  acquiring  its  velocity. 

§55. — If  the  force  become  constant  and  equal  to  F,  we  have  from 
Equation  (18) 

F  _  dh 
M  ~  dfi' 

Multiplying  by  dt  and  integrating,  we  get 

|.,  =  |+e=,+   (7      .     .      (20); 

and  if  the  body  be  moved  from   rest,  the  velocity    will   be  equal  to 
zero  when  t  is  zero ;   whence   C  =  0,  and 

F    * 

Multiplying  Equation  (22)  by  dt,  after  omitting  C  from  it,  and 
integrating  again,  we  find 

F    ^ 

and  if  the  body  start  from  the  origin  of  spaces,  C  will  be  zero,  and 
F   /2 


M    2 


=  « (24). 


46     ELEMENTS  OF  ANALYTICAL  MECHANICS. 

Making    t  equal    to    one   second,  in   Equations   (24)   and  (23),  and 
dividing  the  last  by  the  first,  we  have 


1  -  i- 

2"  "~   -y 


(25). 


That  is  to  say,  when  a  body  is  moved  from  rest  by  the  action  of 
a  constant  force,  the  velocihj  generated  in  the  first  unit  of  time  is 
measured  by  double  the  space  described  in  acquiring  this  velocity. 

g5(5. — The  dynamical  measure  for  the  intensity  of  a  force,  or  the 
pressure  it  is  capable  of  producing,  is  assumed  to  be  the  effect  this 
pressure  can  produce  in  a  unit  of  time,  this  effect  being  a  quantity 
of  motion,  measured  by  the  product  of  the  mass  into  the  velocity 
generated.  This  assumed  measure  must  not  be  confounded  with  the 
quantity  of  work  of  the  force  while  producing  this  effect.  The 
former  is  the  measure  of  a  single  pressure;  the  latter,  this  pressure 
repeated  as  many  times  as  there  are  points  in  the  path  over  which 
this   pressure   is   exerted. 

Thus,  let  the  body  be  moved  from  A  to 
B,  under  the  action  of  a  constant  force,  in 
one  second;  the  velocity  generated  will, 
Equation  (25),  be  2.1^.  Make  BC^2AB, 
and  complete  the  square  BCFE.  BE  will 
be  equal  to  v\  the  intensity  of  the  force 
will  be  M.v;  and  the  quantity  of  work, 
the  product  of  M.v  by  AB,  or  by^  its 
equal  ^  v ;  thus  making  the  quantity  of 
work  ^  M  v"^,  or  the  mass  into  one  half  the 
square  BF;  which  agrees  with  the  result  obtained  from  Equation  (21), 

EQUILIBEIUM. 

^^T,— Equilibrium  is  a  term  employed  to  express  the  state  of 
two  or  more  forces  which  balance  one  another  through  the  interven- 
tion of  some  body  subjected  to  their  simultaneous  action.  When 
applied   to  a  body,  it  means  that  the  body  is  at  rest. 


MECHANICS     OF     SOLIDS.  47 

We  must  be  careful  to  distinguish  between  the  extraneous  forces 
which  act  upon  a  body,  and  the  forces  of  inertia  which  they  may,  or 
may   not,  develop. 

If  a  body  subjected  to  the  simultaneous  action  of  several  extraneous 
forces,  be  at  rest,  or  have  uniform  motion,  the  extraneous  forces  are 
in  equilibrio,  and  the  force  of  inertia  is  not  developed.  If  the  body 
have  varied  motion,  the  extraneous  forces  are  not  in  equilibrio,  but 
develop  forces  of  inertia  which,  with  the  extraneous  forces,  are  iu 
equilibrio.  Forces,  therefore,  including  the  force  of  inertia,  are  ever 
in  equilibrio  ;  and  the  indication  of  the  presence  or  absence  of  the 
force  of  inertia,  in  any  case,  shows  that  the  body  is  or  is  not  chang- 
ing its  condition  in  respect  to  rest  or  motion.  This  is  but  a  conse- 
quence of  the  universal  law  that  every  action  is  accompanied  by  an 
equal  and  contrary  reaction. 

TIIE    CORD. 

§  58. — A  cord  is  a  collection  of  material  points,  so  united  as  to 
form  one  continuous  line.  It  will  be  considered,  in  what  immedi- 
ately follows,  as  perfectly  flexible,  inextensible,  and  without  thickness  or 
weight. 

§  59, — By  the  tension  of  a  cord  is  meant,  the  effort  by  which  any 
two  of  its   adjacent  particles  are  urged  to  separate  from  each  other. 

^  60. — Two    equal    forces,  P    and   P\  applied    at    the    extremities 
^,  ^^  of  a  straight  cord,  an'd 
acting  in  opposite  directions 

from    its    middle    point,  will  ^ -^'  -^ -? 

maintain  each  other  in  equi- 
librio.     For,    all    the    points 

of  the  cord  being  situated  on  the  line  of  direction  of  the  forces,  anv 
one  of  them,  as  0.  may  be  taken  as  the  common  point  of  applica- 
tion without  altering  their  effects ;  but  in  this  case,  the  forces  being 
equal  will,    §o4_,    neutralize    each   other. 


48 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


8  61. — If  two    equal  forces,  P  and  P',  solicit  in  opposite  directions 
the   extremities   of  the   cord 
A  A'',  the  tension  of  the  cord 

will  be  measured  by  the  in-  t: Jt  A p 

tensity  of  one  of  the  forces. 
For,  the    cord   being  in  this 

case  in  equilibrio,  if  we  suppose  any  one  of  its  points  as  0,  to  become 
fixed,  the  equilibrium  will  not  be  disturbed,  while  all  communica- 
tion between  the  forces  will  be  intercepted,  and  either  force  may 
be  destroyed  without  affecting  the  other,  or  the  part  of  the  cord  on 
which  it  acts.  But  if  the  part  ylO  of  the  cord  be  attached  to  a 
fixed  point  at  0,  and  drawn  by  the  force  P  alone,  this  force  must 
measure   the   tension. 

THE    MUFFLE. 


1 62. — Suppose  ^4,  A\  B,  B\  &;c.,  to   be    several    small  wheels  or 
pulleys  perfectly  free 
to  move  about  their 
centres,    which,   con- 
ceive for  the  present  ■^' 
to    be   fixed    points. 
Let  one  end  of  a  cord 
be  fastened  to  a  fixed 
point     (7,     and     be 
wound     around     the 
pulleys  as  represent- 
ed in  the  figure ;   to   the   other   extremity,    attach   a   weight  w.      The 
weight  ?f>  will    be   maintained   in   equilibrio    by    the    resistance  of  the 
fixed  point   (7,  through  the  medium  of  the  cord.     The  tension  of  the 
cord   will    be   the   same    throughout   its   entire   length,    and    equal    to 
the  weight  to ;    for,  the    cord    being  perfectly  flexible,  and    the  wheels 
perfectly    free    to    move    about    their    centres,    there    is    nothing    to 
intercept  the  free  transmission  of  tension  from  one  end  to  the  other. 

Let   the   points  s   and   r  of  the   cord   be  supposed  for    a   moment, 
fixed;  the   intermeriate  portion  sr  may  be  removed  without  affecting 


I 


MECHANICS     OF    SOLIDS.  49 

the  tension  of  the  cord,  or  the  equilibrium  of  the  weight  w.  At 
the  point  r,  apply  in  the  direction  from  r  to  a,  a  force  whose  inten- 
sity is  equal  to  the  tension  of  the  cord,  and  at  s  an  equal  force 
acting  in  the  direction  from  s  to  b;  the  points  r  and  s  may  now  be 
regarded  as  free.  Do  the  same  at  the  points  s',  r\  s^\  r^\  s^''  and 
r''',  and  the  action  of  the  weight  w,  upon  the  pulleys  A  and  A^  will 
be  replaced  by  the  four  forces  at  5,  s',  s'^  and  s'^\  all  of  equal  in- 
tensity and  acting   in    the    same   direction. 

Now,  let  the  centres  of  the  pulleys  A  and  A'  be  firmly  con- 
nected, with  each  other,  and  with  some  other  fixed  point  as  m,  in 
the  direction  of  BA  produced,  and  suppose  the  pulleys  diminished 
indefaiitely,  or  reduced  to  their  centres.  Each  of  the  points  A  and 
A'  will  be  solicited  in  the  same  direction,  and  along  the  same  line, 
by  a  force  equal  to  Sw*,  and  therefore  the  point  m,  by  a  force 
equal   to   4w. 

Had  there  been  six  pulleys  instead  of  four,  the  point  m  would 
have  been  solicited  by  a  force  equal  to  Qw,  and  so  of  a  greater 
number.  That  is  to  say,  the  point  m  would  have  been  solicited  by 
a  force  equal  to  w,  repeated  as   many  times  as  there  are  pulleys. 

If  the  extremity  C  of  the  cord  had  been  connected  with  the  point 
m,  after  passing  round  a  fifth  pulley  at  (7,  the  pomt  in  would 
have  been  subjected  to  the  action  of  a  force  equal  to  5^ ;  if 
seven  pulleys  had  been  employed,  it  would  have  been  urged  by  a 
force  Itv ;  and  it  is  therefore  apparent,  that  the  intensity  of  the 
force  which  solicits  the  point  w,  is  found  by  multiplying  the  tension 
of  the  cord,  or  weight  w,  bg  the  number  of  jndleys. 

This  combination  of  the  cord  with  a  number  of  wheels  or  pulleys, 
is  called  a  muffl.e. 

§  63. — Conceive  the  point  m  to  be  transferred  to  the  position 
in/  or  vi''\  on  the  line  AB.  The  centres  of  the  pulleys  A,  A%  &c., 
being  invariably  connected  with  the  point  m,  will  describe  equal 
paths,  and  each  equal  to  vi  in\  or  m  m^',  so  that  each  of  the  parallel 
portions  of  the  cord  will  be  shortened  in  the  first  case,  or  length- 
ened in  the  second,  by  equal  quantities;  and  if  e  denote  the  length 
of  the   path    described    l)y  vi,    n   the    number  of  parallel  portions  of 


50 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


the  cord,  which  is  equal  to  the  number  of  pulleys,  and  |,  the  change 
in  length  of  the  portion  uio  in  consequence  of  the  motion  of  m, 
we  shall  have,  because  the  entire  length  of  the  cord  remains  the  same, 

w  .  e  =  I (26). 

The  first  member  of  this  equation  we  shall   refer  to  as  the  change 
in  length  of  cord  on  the  pulleys. 

§64. — The  action  of  any  force  P,  upon  a  material  point,  may  be 
replaced  by  that  of  a  mufile,  b>j  making  the  tension  of  its  cord  equal 
to  the  intensity  of  the  given  force,  divided  by  the  number  of  imrallel 
"portions  of  the  cord. 

EQUILIBRIUM    OF    A   RIGID    SYSTEiL 

§  65. — Let  M  represent  a  collection  of  material  points,  united  in 
any  manner  whatever,  forming  a  solid  body,  and  subjected  to  the 
action  of  several  forces,  P,  P',  P'\  P"\  &c.  ;  and  suppose  these 
forces  in  equilibrio. 

Find  the  greatest  force  u\  which  will  divide  each  of  the  given 
forces  without  a  remainder ;    replace  the  force  P  by  a  mvffie,   having 


a  number  of  pulleys   denoted   by  —  ;    the   tension  of  the  cord   will 


MECHAXICS     OF    SOLIDS. 


51 


be  denoted  by  w.  Do  the  same  for  each  of  the  forces,  and  we 
shall  have  as  many  muffles  as  there  are  forces,  and  all  the  cords 
will  have  the  same   tension. 

Let  the  several  cords  be  united  into  one,  as  represented  in  the 
figure,  one  end  being  attached  at  C,  the  other  acted  upon  by  a  weight 
equal  to  the  force  w.  The  action  upon  the  body  will  remain  un- 
changed, that  is,  the  substituted  forces,  including  ?<»,  will  be  in  oqui- 
librio. 

In  this  state  of  the  system,  let  a  force   Q  be   applied  to  put   the 
body   in   motion,   and   at   the    instant   motion    begins,    withdraw    this 
force  and  stop  the  motion  before  the  equilibrium  of  the  forces  is  des- 
troyed.    The  points  of  application  of 
the    original    forces    will    each  have 
described  an  indefinitely  small  path, 
as  m  n.      Let  m  r  be  the  projection 
of  this  path  upon  the  original  direc- 
tion of  the    force,  and    denote   the 
length  of  this  projection  by  e.     Join 
the  point  n  with   any    point    o,   on 
the    direction    of   the    force   and   at 
some  definite  distance  from  m.     From  the  triangle  o/tr,  we  have 


on    =  or    +  nr 


the  displacement  bemg   indefinitely   small,  nr~   may   be   neglected   in 

— 2 

comparison  with  or  ,  being  an  indefinitely  small  quantity  of  the  second 
order ;   hence, 


and. 


om  —  on  =:  ora  —  or  =^  e. 


But  the  number  of  pulleys   in    the    muffle    which   acts    along   the 
direction  of  the   force  P   is. 


P 

w 


hence,  the   change   in   the   length  of  the   cord   on  the  pulleys  of  this 


52  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

muffle,  caused  by  the  slight  motion  of  the  point  of  application  of  the 
force  P,  will,   since  the  centre  of  the  pulley  B  is  fixed,  be 

P.e. 
w 

and  denoting  by  e',  c",  e'",  &c.,  the  projections  of  the  paths  described 
by  the  points  to  which  the  forces  P\  P'\  P"\  &c.,  are  respectively 
applied,  on  the  original  directions  of  these  forces,  we  shall  have 

P'.e'      P".e"      P"'.e"'    • 

)       ?       5     &C., 

WWW 

for  the  corresponding  changes  in  the  length  of  the  cord  on  the  other 
muffles. 

In  all  these  changes,  the  cord  being  inextensible,  its  entire  length 
remains  the  same,  and  if  the  change  in  length  which  the  portion  uw 
undergoes,  be  denoted  by  |,  we  shall  have 

—  (P.  e  +  P'.  e'  +  P".  e"  +  P'".  e'"  +  &c.)  +  1  =  0     .     .  (27.) 

This  equation  expresses  the  algebraic  sum  of  all  the  changes  in 
the  length  of  the  several  parts  of  the  cord,  between  the  points  of 
application,  and  the  fixed  point  towards  which  the  points  of  applica- 
tion are  solicited ;  the  effect  of  these  changes  being  to  shorten  some 
and  lengthen  others,  some  of  the  terms  of  Equation  (27)  must 
be  negative. 

Now  it  is  one  of  the  essential  properties  of  a  system  of  forces 
in  equilibrio,  that  a  body  subjected  to  their  action  is  just  as  free  to 
move  as  though  these  forces  did  not  exist.  The  additional  force  Q, 
therefore,  was  wholly  employed  in  overcoming  the  inertia  of  the 
body  ;  it  was  neither  assisted  nor  opposed  by  the  forces  represented 
by  the  action  of  the  muffles,  because  these  forces  balanced  each 
other,  and  the  motion  was  arrested  before  the  points  of  application 
were  sufficiently  disturbed  to  break  up  the  equilibrium.  But  the 
weight  w,  is  one  of  the  forces  in  equilibrio;  and  the  other  forces 
which    kept   this   weight   from    moving   before   the  application  of  the 


MECHANICS    OF    SOLIDS.  53 

force   Q,   will    keep    it    from   moving   during    the    slight    disturbance. 

We  shall,  therefore,  have 

g=0, 

and  Equation  (27)  will  reduce  to, 

Pe  +  P'e'  +  P"e"  +  P"'e"'  +  &c.  =  0  ;     .     .     .     .   (28). 

g66. — It  may  be  objected,  that  the  given  forces  are  incommensu- 
rable, and  that  therefore,  a  force  cannot  be  found  which  will  divide 
each  without  a  remainder;  to  which  it  is  answered,  that  Equation 
(28),  being  perfectly  independent  of  the  value  of  the  weight  w,  or 
tension  of  the  cord,  this  weight  may  be  taken  so  small  as  to  render 
the  remainder  after  division  in  any  particular  case,  perfectly  inappre- 
ciable. 

8  67. — The  indefinitely    small    paths   m  n,  vi'n',    described    by    the 
points  of  application  of  the  forces,  P  and  P\  during  the  slight  motion 
we  have  supposed,  are  called  virtual  veloci- 
ties ;   and  they  are  so  called,  because,  being 

the    actual    distances    passed    over    by    the  ^/^J 

points  to   which   the   forces   are   applied,  in        ^'L '^ 

the   same   time,   they   measure   the   relative         Jir-^. ^-?' 

rates  of  motion  of  these  points.      The  dis- 
tances rvi  and   r'm\   represented   by  e  and 

e',  are  therefore,  the  projections  of  the  virtual  velocities  upon  the 
directions  of  the  forces.  These  projections  may  fall  on  the  side 
towards  which  the  forces  tend  to  urge  these  points,  or  the  reverse, 
depending  upon  the  direction  of  the  motion  imparted  to  the  system. 
In  the  first  case,  the  projections  are  regarded  as  positive,  and  in  the 
second,  as  negative.  Thus,  in  the  case  taken  for  illustration,  m  r  is 
positive,  and  m'r'  negative.  The  products  Pe  and  P'e',  are  called 
virtual  moments.  They  are  the  elementary  quantities  of  work  of  the 
forces  P  and  P\  The  forces  are  always  regarded  as  positive ;  the 
sign  of  a  virtual  moment  will  therefore  depend  upon  that  of  the 
projection  of  the  virtual   velocity. 

g68. — Referring  to  Equation  (28),  we  conclude,  therefore,  that  xvhen- 
ever  several  forces   are   in  equilihrio,  the  algebraic  sum  of  their  virtual 


64:     ELEMENTS  OF  ANALYTICAL  MECHANICS. 

moments  is  equal  to  zero ;   and  in  this  consists  what  is  called  the  prin- 
ciple of  virtual  velocities. 

§69. — Reciprocally,  if  in  any  system  of  forces,  the  algebraic  sum 
of  the  virtual  moments  be  equal  to  zero,  the  forces  will  be  in  equi- 
Hbrio.  Tor,  if  they  be  not  in  equilibrio,  some,  if  not  all  the  points 
of  application  will  have  a  motion.  Let  q,  q',  q",  &c.,  be  the  pro- 
jections of  the  paths  which  these  points  describe  in  the  first  instant 
of  time,  and  Q,  Q',  Q",  &c.,  the  intensities  of  such  forces  as  will, 
when  applied  to  these  points  in  a  direction  opposite  to  the  actual 
motions,  produce  an  equilibrium.  Then,  by  the  principle  of  virtual 
velocities,  we  shall  have 

Fe  +  F'e'  +  P"e"  +  &c.  +  Qq  +  Q'q'  +  Q"q"  +  &c.  =  0. 
But  by  hypothesis, 

Pe  +  P'e'  +  P"e"  +  &c.  =  0, 
and  hence, 

Qq  +  Q'q'  +  Q"q"  +  &c.  =  0  .    .    .    (28)' 

Now,  the  forces  Q,  Q',  Q'',  &:c.,  have  each  been  applied  in  a  direc- 
tion contrary  to  the  actual  motion ;  hence,  all  the  virtual  moments  in 
Equation  (28)'  will  have  the  negative  sign ;  each  term  must,  therefore, 
be  equal  to  zero,  which  can  only  be  the  case  by  making  Q,  Q\  Q"', 
(fee,  separately  equal  to  zero,  since  by  supposition  the  quantities 
denoted  by  q,  q\  q",  are  not  so.  We  therefore  conclude,  that  when 
the  algebraic  sum  of  the  virtual  moments  of  a  system  of  forces  is 
equal  to  zero,  the  forces  will  be  in  equilibrio. 

Whatever  be  its  nature,  the  effect  of  a  force  will  be  the  same  if 
we  attribute  its  effort  to  attraction  between  its  point  of  application 
and  some  remote  point  assumed  arbitrarily  and  as  fixed  upon  its  line 
of  direction,  the  intensity  of  the  attraction  being  equal  to  that  of  the 
force.  Denote  the  distance  from  the  point  of  application  of  P,  to 
that  towards  which  it  is  attracted  by  p,  and  the  corresponding  dis- 
tances in  the  case  of  the  forces  P',  P",  &c.,  by  p',  p",  &c.,  respect- 
ively ;  also,  let  5p,  Sp\  Sp",  &c.,  represent  the  augmentation  or  dimi- 
nution of  these  distances  caused  by  the  displacement,  supposed  indefi- 
nitely small,  then  §  65,  will 

e  =  Sp,  e'  =  8p\  e"  =  Sp'\  &c., 


MECHANICS     OF    SOLIDS.  55 

and  Equation  (28)  may  be  written  . 

P8p  +  P'Sp'  +  P"^p"  +  &c.  =  0     .     .     .  (20), 

in  Avliich  the  Greek  letter  5  simply  denotes  change  in  the  value  of 
the  letter  written  immediately  after  it,  this  change  arising  from  the 
small   displacement. 

§70. — If  the  extraneous  forces  applied  to  a  body  be  not  in  equi- 
librio,  they  will  communicate  motion  to  it,  and  will  develop  forces  of 
inertia  in  its  various  elementary  masses  with  which  they  will  be  in 
equilibrio  ;  and  if  extraneous  forces  equal  in  all  respects  to  these  forces 
of  inertia  were  introduced  into  the  system,  the  algebraic  sum  of  the 
virtual  moments  would  be  equal  to  zero. 

But  if  m  denote  the  mass  of  any  element  of  the  body,  s  the 
path  it  describes,  its  force  of  inertia  will,  Eq.  (13),  be 

and  denoting  the  projection  of  its  virtual  velocity  on  s  by  (35,  its  vir- 
tual moment  will  be 

and  because  the  forces  of  inertia  act  in  opposition  to  the  extraneous 
forces,  their  virtual  moments  must  have  signs  contrary  to  those  of 
the  latter,  and  Equation  (29)  may  be  written 

2P.  (J»  _  2w  .  ^'  .  5s  =:  0 :  .     .     .     .  (30). 
^      dfi 

in  which  2  denotes  the  algebraic  sum  of  the  terms  similar  to  that 
written  immediately  after  it. 

PRINCIPLE    OF    D'ALEMBERT. 

§71. — This  simple  equation  involves  the  whole  doctrine  of  Mechanics. 
The  extraneous  forces  P,  P',  P",  &c.,  are  called  impressed  forces. 
The  forces  of  inertia  which  they  develop  may  or  may  not  be  equal  to 
them,  depending  upon  the  manner  of  their  application.  If  the  impressed 
forces  be  in  equilibrio,  for  instance,  they  will  develop  no  force  of  inertia ; 


66  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

but  in  all  cases,  the  forces  of  inertia  actually  developed  will  be  equal 
and  contrary  to  so  much  of  the  impressed  forces  as  determines  the 
change  of  motion.  The  portions  of  the  impressed  forces  which  deter- 
mine a  change  of  motion  are  called  effective  forces  ;  and  from  Equation 
(30),  we  infer  that  the  impressed  and  effective  forces  are  always  in  equi- 
librio  when  the  directions  of  the  latter  are  reversed,  and  will  pre- 
vent all  change  of  motion.  This  is  usually  known  as  D'Alemberi's 
Frincrple,  and  is  nothing  more  than  a  plain  consequence  of  the  law 
that  action  and  reaction  are  ever  equal  and  contrary. 

This  same  principle  is  also  enunciated  in  another  way.  Since  the 
effective  forces  reversed  would  maintain  the  impressed  forces  in  equi- 
librio,  and  prevent  them  from  producing  a  change  of  motion^  it 
follows  that  ivhatever  forces  may  be  lost  and  gained  must  he  in  equili- 
hrio ;  else  a  motion  different  from  that  which  actually  takes  place 
must  occur,  a  supposition  which  it  were  absurd  to  make. 

§  72.— Equation  (30),  is  of  a  form  too  general  for  easy  discussion. 
To  transform  it,  refer  the  directions  of  the  forces  and  their  points 
of  application  to  three  rectangular  axes. 

Denote  by  a,  /3,  7,  the  angles  which  the  direction  of  the  force 
P  makes  with  the  axes  x,  y,  z,  respectively  ;  by  a,  5,  c,  the  angles 
which  its  virtual  velocity  makes  with  the  same  axes;  and  by  9,  the 
angle  which  the  virtual  velocity  and  direction  of  the  force  make  with 
each  other,  then  will 

cos  9  =  COS  a  .  COS  a  +  cos  b  .  cos  (3  -f  cos  c  .  cos  7. 
Denote  by  k,  the  virtual  velocity,  and  multiply  the   above    equation 
by  Fk,  and  we  have 

Pk  cos  <p  =  Pk  cos  a  .  cos  a  +  Pk  cos  b  .  cos  (3  +  Pk  cos  c .  cos  7 ; 
But   denoting    the   co-ordinates  of  the   point   of  application  of  P  by 
X,  y,  z,  we  have 

k  cos  (p  =  5p',    k  cos  a  =  Sx  ;   k  cos  b  =  Si/  ;   k  cos  c  =  Sz  ; 
and  these  values  substituted  above,  give 

P. Sp  =  P  cos  a. Sx  +  P  cos  ^.§1/  -{-  P  cosy. Sz.    .    .  (31). 
Similar  values  may  be  found  for  the  virtual  moments  of  other  forces. 


MECHANICS    OF    SOLIDS.  57 

§  73. — Again 

ds^  =  dx~  +  di/  +  ^z^  ; 

differentiating  and  multiplying  by  m  •  ^^^r-^  wc  have 

dh    ,  d-^x   dx  d-y  dij  d'-z    dz 

rf^2  dt^    ds  dl-    ds  dt^    ds 

and  denoting  by  8x,  Stj,  5z,  the  projections  of  Ss  on  x,  y,  0,  respectively, 
we  have 

^.Ss  =  Sx',   ^-l.Ss^Sy,    ^±.Ss  =  6z, 
ds  ds  ds 

whence, 

d^s      ^  d^x      ^      ,  dhj  ^^^     X  /Qo\ 

and  similar  expressions  may  be  found  for  the  virtual  moments  of  the 
forces  of  inertia  of  the  other  elementary  masses. 

§  74. — If  the  intensity  of  the  force  P,  be  represented  by  a  portion 
of  its  line  of  direction,  which  is  the  practice  in  all  geometrical 
illustrations  of  Mechanics,  the  factors  P  cos  a,  P  cos  ^,  and  P  cos  y, 
in  Equation  (31),  would  represent  the  intensities  of  forces  equal  to 
the  projections  of  the  intensity  P,  on  the  axes  ;  and  regarding  these 
as  acting  in  the  directions  of  the  axes,  the  factors  Sx,  8rj^  and  fe,  will 
represent  their  virtual  velocities,  which  virtual  velocities  will  coincide 
with  their  own  projections. 

Again,  Equation  (32), 

d'^x  d^u  d^z 

■^•-rf^'    ^'-S^'    ^'7^' 

are  forces  of  inertia  in  the  directions  of  the  axes,  and  Sx,  Sy,  Sz,  are 
their  virtual  velocities  ;  these  also  coincide  with  their  own  projcc 
tions. 

The  values  of  these  virtual  velocities  depend  upon  the  nature  of 
the  motion. 


58 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


FREE    MOTION. 


§75. — A  body  is  said  to  have  free  motion,  when  it  pursues  the 
path  and  takes  the  velocity  due  to  the  directions  and  intensities  of 
the  extraneous  and  active  forces  impressed  upon  it.  This  motion 
is  to  be  distinguished  from  that  in  which  the  body  is  constrained, 
by  the  interposition  of  some  rigid  surface  or  line,  to  take  a  path 
different  from  that  which  it  would  describe  but  for  such  interposi- 
tion. A  body  simply  falling  under  the  action  of  its  own  weight  is 
a  case  of  free  motion.  The  same  body  rolling  down  an  inclined 
surface,  is  not. 

The  most  general  motion  we  can  attribute  to  a  body  is  one 
of  translation  and  of  rotation  combined.  A  motion  of  transla- 
tion carries  a  body  from  place  to  place  through  space,  and  its 
position,  at  any  instant,  is  determined  by  that  of  some  one  of  its 
elements.  A  motion  of  rotation  carries  the  elements  of  a  body 
around  some  assumed 
point.  In  this  investi- 
gation, let  this  point 
be  that  which  deter- 
mines the  body's  place. 

Denote  its  co-ordi. 
nates  by  x^  y,  z,  and 
those  of  the  element 
?«,  referred  to  this  point 
as  an  origin  by  x\  y\ 
z' ;  there  will  thus  be 
two  sets  of  axes,  and 
supposing  them  parallel, 
we  have 


/. 


2'\ 


771" 


-X 


and  differentiating, 


dx  =.  dx^  -\-  dx', 
dy  =  dy,  +  dy', 
dz  =z  dz^  -\-  dz'. 


(33), 


(34). 


MECHANICS    OF    SOLIDS. 


5D 


Demit  from  7?i,  the  per- 
pendiculars mX',  m  y,  mZ' 
upon  the  movable  axes. 
Denote  the  first  by  r',  the 
second  by  r",  and  the  third 
by  ?•'".  Let  0',  0",  0"\ 
be  the  projections  of  7n,  on 
the  planes  xy,  xz,  y  z,  res- 
pectively. Join  the  several 
points  by  right  lines  as 
indicated  in  the  figure. 

Denote   the   angle 


z 

/ 

/ 

07: 

^^ -f 

\hC^ 

o> 


Then   will 
the  triangle  m  Z'  0"  give 


m  Z'  0"  by  9, 
m  X'  0'  by  -us, 
m  Y'  0'"  by  4.. 


the  triangle  m  Y'  0 


■".    1: 


f    y'  =1  r'  cos  •cr,  \ 
the  triangle  mX'  0\  \     ,         ,    .  \ 


r'"  cos  (p 
r'"  sin  9 


;l 


=  r"  sin  %j-',  J 
=  r"  cos  4',  f 


(35), 
(30), 
(37). 


We  here  have  two  values  of  x',  one  dependent  upon  9,  and  the 
other  upon  ■\^.  Hence,  if  we  differentiate  r',  supposing  9  variable 
and  4^  constant,  r'"  will  also  be  constant,  since  the  point  m  will 
describe  the  arc  of  a  circle  about  the  axis  z.     Whence, 


dx' 


r'"  sin  9  .  c?9 


differentiating  x',  supposing  4^  variable,  and  9  constant,  r"  will  also  be 

constant,  since  m  will   describe   the    arc  of  a  circle   about  the  axis  y ; 

whence, 

dx'  =  r"  cos^.d-^. 

These    being    the    partial    differentials  of  x\   we    have    for   the   total 
differential, 

dx'  =  r"  cos  4^  •  f?  4^  ~  ^"'  sii'i  9  •  <^9> 


CO 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


replacing  r"  cos  -v^  and  r'"  sin  9,   by  their   values  in  the  above  Equa- 
tions, and  we  get 

dx'  =z  z'  .d-\j  —  y'  .d:p; 

and  in  the  same  way,  I    .     .     .  (38) 

dy'  =  x' .  dtp  —  z' .  d  zi, 
d  z'  =  y' .  d-a  —  x'  .d  ■\^, 

which  substituted  in  Equations  (34),  give 


dx  =  dXj  -\-  z'  .d-\^  —  y' '  d(p, 
dy  ^=  dy^  -\-  x'  .d  9  —  z' .  dzi, 
dz  =  d  Zj  +  y'  .d  -m  —  x' .d\. 


•  (39), 


and  because  the  displacement  is  indefinitely  small,  we  may  write 

.     .     .(39)' 


^x  —  8x^  +  z' .5-]^  —  y'  .S(p, 
Sy  =  S  y^  +  x' .  5  (p  —  z'  .Szi, 
Sz   =z  Szj  -\-  y'  .$■&  —  x'  .S-],; 


and  tnese  in  Equations  (31)  and  (32),  give 

'  P  cos  a .  Sx^  -\-  P  cos  ^  .5y^  -\-P  cos  y .  Sz^ 
^      _    J    +  -^  •  C^'  •  cos  [3  —  y' .  cos  a) .  Sep 
+  P  .  {z' .  cos  a  —  x' .  cos  y)  .  S-^i 
^  +  P  .{y' .  cos  y  —  z' .  cos  /3) .  Ssi. 


d^x  d"  V  d  z 

m  •  -^rir  '  Sx^  +  m  '  -pr-  'Syj  +  m  •  -^r;  •  Sz 


dt^    ""'    '    ■"    df 

x' .  d?y  —  y' .  ^x 


dfi 


-\-  m 


m '  —j-r  'Ss  —\ 


df 


Sep 


z' .  d'^x  —  x' .  d'^z     y , 
+  m.. ^, 54^ 


d'^s' 


Similar  values  may  be  found  for  P' .  8p'  and  «i' .  —r-^  •  Ss\  &c.     In 

these  values  Sx^^  Sy^,  and  S  z^ ,  will  be  the  same,  as  also  S(p,  5-^,  and 
Szi,  for  the  first  relate  to  the  movable  origin,  and  the  latter  to  the 
angular  rotation  which,  since  the  body  is   a  solid,  must  be  of  equal 


MECHANICS     OF     SOLIDS, 


Gl 


values  for  all  the  elements ;  so  that  to  find  the  values  of  the  virtual 
moments  of  the  other  forces,  it  will  be  only  necessary  suitably  to 
accent  P,   a,  /3,  7,    x,  y,  z,    x\  y\  z'. 

These   values   being   found   and   substituted   in   Equation   (30),   we 
shall  find, 


2  m  '  —r-:r  )  OX^ 


(2  P.  cos  a        _..„     ^^^ 
+  (^2P.cos/3-2m.^|-)5y, 
+  (1  F. cos  y-:Em~)  8z, 
+       2P.  (x^cos/S— /.cosa)-i:7?i-       •     ^.o       J  <^9 

+      2P.  (s'.cosa— a;'.  COS7)— 2  m -j^ J  04. 

r           ,  ,                  ,          n^              y'  .dH  —  z' .  cZ2y-| 
+   [_2P.  (y'.cosy-3'.cos/3)-2,7z.^^ -^^ ^J  h 


^-0.(40) 


■ztf 


But  the  displacement  being  entirely  arbitrary,  the  least  considera- 
tion will  show  that  ox^,  ^ y ,^  5z^,  Sep,  S ■]^,  and  Szi,  are  Avholly  inde- 
pendent of  each  other,  and  this  being  the  case,  the  principle  of  inde- 
terminate co-efficients  requires  that 


2  P .  cos  a  —  2  in  •  -— -  =  0, 

2  P .  cos  /3  —2m.  -y-f  =0,     y 
'^  dfi  '     ' 

2P.C0S7  -2m. ^.=  0; 


{A). 


2  P.  (x' .  COS  /3  —  y' .  COS  a)  —  2  ?;i '■^j:r =  ^5 


2  P .  (2' .  cos  a  —  a;' .  cos  7)  —  2 : 


dt^ 

c' 

d'"X  -  x' 

.d'^z 

dt'^ 

y' 

.d" 

z  —  z' 

dhj 

=  0, 


2  P .  (//' .  cos  7  —  2' .  cos /3)  —  2  m  •  "  ,^ ~  '"  "    =  0. 


.     .   (5). 


62  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

§76. — These  six  equations  express  either  all  the  circumstances  of 
motion  attending  the  action  of  forces,  or  all  the  circumstances  of 
equilibrium  of  the  forces,  according  as  inertia  is  or  is  not  brought 
into  action;  and  the  study  of  the  principles  of  Mechanics  is  little 
else  than  an  attentive  consideration  of  the  conclusions  which  follow 
from  their  discussion. 

Equations  (^1)  relate  to  a  motion  of  translation,  and  Equations 
(J5)  to  a  motion  of  rotation.  They  are  perfectly  symmetrical  and 
may  be  memorized  with  great  ease. 

COMPOSITION    AND    RESOLUTION    OF    FOECES. 

§77. — When  a  body  is  subjected  to  the  simultaneous  action  of 
several  extraneous  forces  which  are  not  in  equilibrio,  it  will  be  put 
in  motion  ;  and  if  this  motion  may  be  produced  by  the  action  of 
a  single  force,  this  force  is  called  the  resultant,  and  the  several  forces 
are    termed  com])onents. 

The  resultant  of  several  forces  is  a  single  force  which,,  acting  alone, 
will  produce  the  same  effect  as  the  several  forces  acting  simultaneously ; 
and  the  cotnponents  of  a  single  force,  are  several  forces  whose  simulta- 
neous action  j^roduces    the   same   effect   as    the   single  force. 

If,  then,  several  extraneous  forces  applied  to  a  body,  be  not  in 
equilibrio,  but  have  a  resultant,  a  single  force,  equal  in  intensity  to 
this  resultant,  and  applied  so  as  to  be  immediately  opposed  to  it, 
will  produce  an  equilibrium,  or  what  amounts  to  the  same  thing, 
if  in  any  system  of  extraneous  forces  in  equilibrio,  the  resultant  of  all 
the  forces  but  one  be  found,  this  resultant  will  be  equal  in  intensity 
and  immediately  opposed  to  the  remaining  force  ;  otherwise  the  sys- 
tem could   not   be    in  equilibrio. 

Conceive  a  system  of  extraneous  forces,  not  in  equilibrio,  and 
applied  to  a  solid  body,  and  suppose  that  the  equilibrium  may  be 
produced  by  the  introduction  of  an  additional  extraneous  force. 
Denote  the  intensity  of  this  force  by  li,  the  angles  which  its  direc- 
tion makes  with  the  axes  x,  y,  and  z,  by  a,  b,  and  c,  respectively, 
and  the  co-ordinates  of  its  point  of  application  by  x,  y,  z.  Then, 
because   the   inertia  cannot  act,  d-x,  d'^g,  d-z  will  be   zero,  and  taking 


MECHANICS     OF    SOLIDS.  63 

the   two    origins   to    coincide,  Equations  (.1)  and  (/?),  will  give 

i2  cos  a  +  P'  cos  a'  +  P"  cos  a"  +  P'"  cos  a'"  +  &c.  =  0, 
i?  cos  6  +  P'  cos  /3'  +  P"  cos  /3"  +  P'"  cos  /3'"  +  &c.  =  0, 
i2  cos  c  +  P'  cos  y'  +  P"  cos  y"  +  P'"-  cos  7'"  +  &c.  =  0  ; 


^  (a:  cos  b  —  y  cos  a)  +  P'  (x'  cos  ^'  —  y'  cos  a') 
+  P"  (.c"  cos  /3"  -  y"  cos  a")  +  &c. 

i2  (0  cos  a  —  X  cos  c)  +  P'  (2'  cos  a'  —  a;'  cos  y') 
+  P"  {z"  cos  a"  —  x"  cos  y")  +  &c. 

E  {y  cos  c  —  z  cos  ^)  +  P'  (y'  cos  7'  —  s'  cos  /3') 
+  P"  (y"  cos  7"  -  z"  cos  /3")  +  &c. 


f 


=  0. 


Now  P  is  equal  in  intensity  to  the  resultant  of  all  the  other 
forces  of  the  system,  or  in  other  words,  to  the  resultant  of  all  the 
original  forces ;  and  if  we  give  it  a  direction  directly  opposite  to 
that  in  which  it  is  supposed  to  act  in  the  above  equations,  it  be- 
comes in  all  respects  the  same  as  that  resultant,  being  equal  to  it 
m  intensity  and  having  the  same  point  of  application  and  line  of 
direction.  Adding,  therefore,  180°  to  each  of  the  angles  a,  b,  and  c, 
the  first  terms  of  the  foregoing  equations  become  negative,  and 
transposing  the  other  terms  to  the  second  member  and  changing  all 
the   signs,  we   have. 


P  cos  a  =  P'  cos  a'  +  P"  cos  a"  +  P'"  cos  a'"  +  &c.  =  X; ' 
Rcosb  =  P'  cos  /3'  +  P"  cos  /3"  +  P'"  cos  (3'"  +  &c.  =  V: 
P  cos  c  =  P'  cos  7'  +  P"  cos  y"  +  P'"  cos  y'"  +  «Scc.  =   Z.  . 


(41) 


R  [x  cos  b  —  y  cos  a)  =  < 


P'  {x'  cos  13'  —  y'  cos  a') 
4-P"  (a;"  cos /3"  -  y"  cos  a") 
14-&C. 

r     P'  (s'  cos  a'  -  x'  cos  7') 
P  (2  cos  a  —  a;  cos  c)  =  ^  +P"  (2"  cos  a"  —  x"  cos  7") 

U&c. 


P  (y  cos  c  —  2;  cos  i)  =  < 


P'  (y'  cos  7'  -  2'  cos  /3')     ^ 
+  P"(y"  cos  7"-  z"cos^") 
+  &c. 


3f- 


=K 


(42) 


64:  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


R  cos  a  =:  JT, 

i?  cos  6  =  r,   }- (43) 

R  cos  c  =  Z. 

R  {x  cos  h  —  y  cos  a)  =  Z, 

i?  (s  cos  a  —  a;  cos  c)  =  J/,   }► (44) 

R  (y  cos  c  —  z  cos  5)  =  iV. 

Eliminating  i2  cos  a,  R  cos  6  and  R  cos  c,  from  Equations  (44), 
by  means  of  Equations  (43),  we  get,  by  transposing  all  the  terms  to 
the  first  member, 


Xy  -  Yx  -{-  L  =0,'' 
Zx  -.Xz  +  J/=  0, 
Yz  -  Zy  +  N  =  0. 


(45) 


These  are  the  equations  of  a  right  line.  But  x,  y  and  z  are  the 
co-ordinates  of  the  point  of  application  of  the  resultant;  they  are, 
therefore,  the  equations  of  the  line  of  direction  of  the  resultant  R, 
and  hence  the  point  of  application  of  the  resultant  may  be  taken 
anywhere  on  this  line  without  changing  its  effect.  Any  condition, 
therefore,  expressive  of  the  simultaneous  existence  of  these  equations, 
will  also  express  the  existence  of  this  single  line,  and  of  a  single 
resultant  to   the   system  of  forces. 

|78._To  find  this  condition,  multiply  the  first  of  these  Equations 
by  Z,  the  second  by  F,  the  third  by  X,  and  add^  the  products ; 
we   obtain, 

ZL^-YM+XN=0 (4G). 

g79. — Having  ascertained,  by  the  verification  of  this  Equation, 
that  the  forces  have  a  single  resultant,  its  intensity,  direction,  and 
the  equations  of  its  direction  may  be  readily  found  from  Equations 
(43)  and  (44). 

Squaring   each  of  the   group  (43),  and   adding,  we   obtain, 

R^  (cos2  a  -f  cos2  I  _|_  cos2  ^^  _  x^  +  F2  +  Z2. 


MECHANICS    OF    SOLIDS.  65 

Extracting   the   square   root    and   reducing   by  the   relation, 

cos^  a  +  cos^  b  +  cos^  c  =  1, 
there  will  result, 

R  =  VX2+  ^2  +  22 (47), 

which   gives   the   intensity   of  the   resultant,    since   JT,    Y  and  Z  are 
known. 

Again,  from  the  same  Ec[uations, 

X 

Y 
R' 
Z 

cos   C  =z   — • 

R 

which  make  known   the   direction  of  the   resultant. 
The  group  of  Equations  (44)  give, 

Yx  —  Xy  -  ^P'  (cos  /3'  x'  —  cos  a'  y')  =  0,1 
Zx  -  Xz  ~  -L  P'  (cos  y'  x'  -  cos  a'  z')  =  0,  I  • 
Yz  -  Zy  -  2P'  (cos/3'  g'  -  cos  7' y')  =  O.j 

which   are   the   equations  of  the    direction  of  the  resultant. 


cos  a  = 


cos  h 


(48) 


(49) 


PAEALLELOGKAM    OF   FOKCES. 

§80. — If  all   the  forces  be  applied  to  the   same  point,  this  point 
may  be   taken  as   the   origin  of  co-ordinates,  in  Avhich  case, 

x'  =  x"  =  x'"  &c.  =  0, 
y'  =  y"  =  y'"  &c.  =  0, 
z'  =  z"  =  z'"  &c.  =  0, 

and   the  last   term  in   each  of  Equations    (49),  will   reduce   to   zero. 
Hence,  to   determine   the   intensity,   direction   and   equations    of    the 

5 


66 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


line   of    direction    of   the   resultant,   we    have,    Equations    (47),   (48) 
and  (49), 

B  =  y'X2  +  Z2  +  Z2    .     .    . 


X    ^ 
cos  a  =  — > 

,      r 

cos  6  =  — ) 

Z 

cos  C  =  —  J 
it 


(50) 


(51) 


Yx  -  Xy  =  0, 
Za;  —  Xz  =  0, 
r^  -  Zy  =  oj 


(52) 


The  last  three  equations  show  that  the  direction  of  the  resultant 
passes  through  the  common  point  of  application  of  all  the  forces, 
which  might  have  been   anticipated. 

§81. — Let  the  forces  be  now  reduced  to  two,  and  take  the  plane 
of  these  forces  as   that  of  a;y;   then  will 

y'  =  y"  =  7'"  =  &C.   =  90°  ',   Z  =  0, 

the   last   Equation  of  group  (43)  reduces   to, 

Z  =  0; 
and  the   above  Equations  become, 

B  =  ^X-'  +  y 
X 


cos  a  =  —1 
si 


cos  b 


r 


(53) 
(54) 


cos  c  =  0, 

Tx  -  Xy  =  0 (55) 

The  last  is  an  equation  of  a  right  line  passing  through  the 
origin.  The  direction  of  the  resultant  will,  therefore,  pass  through  the 
point  of  application  of  the  forces.  The  cos  c  being  zero,  c  is  90°, 
and  the  direction  of  the  resultant  is  therefore  in  the  plane  of  the  forces. 


MECHANICS    OF    SOLIDS. 


67 


Substituting  in    Equation   (53),   for  X  and    F,    their   values    from 
Equations  (41),  we  obtain, 

R  =  V  (^'  cos  a'  +  F"  cos  a")2  +  (P'  cos  /3'  +  F"  cos  j8")2 ; 
and  since 


cos2  a'    +  cos2  ^'    —  1^ 
cos2  a"  +  cos2  /3"  =  1, 

this   reduces   to 


E  =  V/"2  +  i^"2  +  2  P'  i^"  (cos  a'  cos  a"  +  cos  /3'  cos  /3")  j 

denoting  the  angle  made  by  the  directions  of  the  forces  by  8,  we 
have, 

cos  a'  cos  a"  +  cos  ^'  cos  /3"  =  cos  S ; 

and   therefore, 

i2  =  ^F'-'  +  P"2  +  2  P'P"  cos  5     ....     (5G) 

from  which  we  conclude  that  the  intensity  of  the  resultant  is  equal 
to  thai  diagonal  of  a  parallelogram  whose  adjacent  sides  represent  the 
directions  and  intensities  of  the  co7nponents,  tvhich  passes  through  the 
point  of  ajyplication. 

§  82. — Substituting  in  Equations  (54),  the  values  of  X  and  Y,  from 
Equations  (41),  we  have, 

R  cos  a  =  F'  cos  a'  +  F"  cos  a", 
i2  cos  6  =  F'  cos  /3'  +  F"  cos  ^", 
and  because 

a'   =  90°  —  /3', 

a"  =  90°  -  /3", 

a     =  90°  -  6, 

these   Equations  reduce   to, 

R  cos  a  ^^  F'  cos  a'  +  P"  cos  a", 
i2  sin  a  =  P'  sin  a'  +  P"  sin  a" ; 


68  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

by  transposing   and   squaring,  we   obtain, 

P"2  cos2  o>i  —  Ji2  cos2  a  —  2  BF'  cos  a  cos  a'  +  P'2  ^ogZ  a', 
P"2  sin2  a"  =:  i22  siii2  a  ~  2  EF'  sin  a  sin  a'  +  F'^  sin2  a'; 

adding   and  reducing, 

P"2  ^  i22  ^_  p'2  _  2EF'  cos  (a  -  a')  ; 
but,  ^ 

a  —  a.'  z=  the  angle  RinF'  =  <p' ; 

hence,  by  transposition  and  reduction, 

i22  -f-  P'2  _  p/'2 

^"^^    =  2EF' ' 

or, 

1     coc.--2.in2^          F'--{R-Fr  _  jF'' +  R -F')iF" +  F'-E) 
1     cos<p_2sm29-  .^^p,         -  ^^y, 

whence,  making 

E  +  F'  -j-  F"        „ 

2 "='^' 

we   obtain, 


'=V^^^I^' (-) 


sin  ^  (p 


from  which  we  see  that  the  direction  of  the  resultant  coincides  with 
the  diagonal  of  the  parallelogram  described  on  the  lines  represent- 
ing  the   intensities   and   directions  of  the   forces. 

Thus,  the  resultant  of  any  two  forces,  applied  to  the  same  material 
point,  is  represented,  in  intensity  and  direction,  by  that  diagonal  of  a 
parallelogram,  constructed  vpon  the  sides  representing  the  intensities 
and  directions  of  the  two  components,  which  passes  through  the  point 
of  application. 

§83. — In  the  triangle  RmF',  since  F'  R  is  equal  and  parallel  to 
the  line  which  represents  the  force  F",  the  angle  mP'R  =  (p,  is  the 
supplement  of  the  angle  8,  made  by  the  directions  of  the  components, 
and  there  will  result  the  following  equation, 

sm  1  (J  =  sm  i(p  =  \/  ^ -^rj;j, ;  •  •  (58) 


MECHANICS     OF    SOLIDS. 


69 


Equation  (57),  will  make  known  the  angle  made  by  the  direction 
of  the  resultant  with  that  of  either  of  two  oblique  components,  pro- 
vided, the  intensities  of  the  components  and  resultant  be  known. 

§  84. — Also,  from  the  two  triangles  R  m  F'   and   R  m  £",   we  find, 
.       .       P".sin(J        1 


sm  (p    = 


sm  9 


R 

.  sin  8 
'R 


(59), 


from  which  the  angles  made 
by  the  direction  of  the  result- 
ant with  its  two  components  may  be  found. 

§85. — Let  there  now  be  the  three   forces   F,  P\   P",    applied    to 
the     material    point    m,    in 
the     directions    m  P,    m  P\ 
mP'\  not  in  the  same  plane  ; 
the  resultant   will  be   repre- 
sented in  intensity  and  direc- 
tion  by    the   diagonal   of   a 
parallelopipedon,  constructed 
upon  the   lines    representing 
the  directions  and  intensities 
of  these   components.      For, 
lay    off   the    distances    in  A, 
m  C,  and    m  B,   proportional 
to  the  intensities  of  the  com- 
ponents   which   act   in   the   direction  of   these  lines,  and  construct  the 
parallelopipedon   £B ;    the   resultant   of    the   components   P'  and   P 
will,    §  82,    be    represented  by  the  diagonal  m  B,  of  the  parallelogram 
mABC;    and  the  resultant  of  this  resultant  and  the  remaining  com- 
ponent P",  will  be  represented  by  the  diagonal  m  D  of  the  parallelo- 
gram  U  711  BjD,    which   is   that  of  the  parallelopipedon. 

§86. — If  the  forces  act  at  right  angles  to  each  other,  the  parallel- 
opipedon will  become  rectangular,  and  the  intensity  of  the  resultant, 
denoted  by  i2,  will  become  known  from  the  formula 


70     ELEMENTS  OF  ANALYTICAL  MECHANICS. 


and  if  the  angles  which  the 
direction  of  the  resultant 
makes  with  those  of  the 
forces  P,  P'  and  P",  be 
represented  by  «,  i,  and  c, 
respectively,  then  ^vdll 

B  cos  a  z=  P, 
Bcosb  =  P', 
Bcosc  =  P". 


Let  three  lines  be  drawn  through  the  point  of  application   m\   of 
the  force  P\  parallel  to  any  three  rectangular  axes  x,y,z;  and  denote 
by  a', /3',  7',  the  angles  which 
the    direction   of   this    force  ^ 

makes   with  these   axes   res- 
pectively ;   then  will 


P'  cos  a', 
P'  cos  /3', 
P'  cos  7', 


A 


-X 


be  the  components  of  the  force  P\  in  the  direction  of  the  axes,  and 
they  will  act  along  the  lines  drawai  through  the  point  m'.  These  are 
the  same  as  the  terms  composing  in  part  Equations  (A),  and  as  the 
effect  of  the  components  is  identical  with  that  of  the  resultant,  these 
components   may  always   be   substituted  for  the  force  P'.      The  same 

for  the  forces  of  inertia,  and   -^-r-,    --r->    and   — r-^ 

ponents  of  this  force  in  the  directions  of  the  axes. 


denote   the  com- 


MECHANICS    OF    SOLIDS. 


71 


§87. — Examples. — 1.  Lot  the  point  in,  be  solicited  by  two  forces 
whose  intensities  are  9  and  5,  and  whose  directions 
make  an  angle  with  each  other  of  57°  30'.  Re- 
quired the  intensity  of  the  force  by  which  the 
point  is  urged,  and  the  direction  in  which  it  is 
compelled  to  move. 

Fii-st,  the  intensity  ;   make  in  Equation  (5G), 

P'   =  9, 
P"  =  5, 
5  =  57°  30'  ; 

and  there  will  result, 


R  =  V  81  +  25  +  90  X  0,  537  =  12,422. 

Again,  substituting   the   values  of  5,  P'  P"  and  R  in  the  first  of 
Equation  (59),  we  have, 

5  X  sin  57°  30' 


sm  9    = 


12.422 


or. 


9'  =  19°  50'  35"  nearly, 


which  is  the  angle  made  by  the  direction  of  the  force  9  with  that  of 
the  resultant. 

2. — Required  the  angle  under  which  two  equal  components  should 
act,  in  order  that  their  resultant  shall  be  the  7i'*  part  of  cither  of  them 
separately. 

Bv  condition,  we  have 


hence. 


p,  ^  p»  ^  ji 


=  S  = 


P'  =  P"  =  nR ; 

nR  -{-  nR  -\-  R        {In  +  \)  R 


and,  Equation  (57), 


•    ,    ,          /{S  -  P')  {S-  P") 
sm  \<p'  =  \J  ^ -^ry,! '^ 


72  ELEMENTS    OF    ANALYTICAL    MECHANICS, 

which  reduces  to 


sin  I- 9'  =  ±  — 


]_ 

2n 


If  n  be  equal  to  unity,  or  the  resultant  be  equal  to  either  force, 

(p  =  60°, 
and,  §83,  the  angle  of  the  components  should  be  120°. 

3, — Kequired  to  resolve  the  force  18  =  a,  into  two  components 
whose  difference  shall  he  5  =  b,  and  whose  directions  make  with 
each  other  an  angle  of  38°  =  S.  Also,  to  find  the  angle  wliich  the 
direction  of  each  component  makes  with  that  of  the  resultant. 

Writing  a  for  E   in  Equation  (53),  we  have, 

P'2  ^  p"2  _(_  2  P'  F"  cos  S  =  a?, 

and  by  condition, 

P'-P"  =  b (c). 

Squaring  the  second  and  subtracting  it  from  the  first,  we  get 

2FP"  (1  +  cos  (J)  =  a2  -  62  J 

which,  replacing  (1  +  cos  B)  by  2  cos^  ^  S,  reduces  to 

a2  —  62 

COS''  f  0 

This  added  to  the  square  of  the  Equation  (c),  gives 


from  which  and  Equation  (c)  we  finally  obtain, 


which  are  the  required  components. 

To  find  the  angles  which  their  directions  make  with  the  resultant, 
we  have  from  Equations  (59), 

9"  =  24°  =  the  angle  which  P"  makes  with  the  resultant. 


MECHANICS    OF    SOLIDS. 


73 


and, 


9'  =  14°  =  angle  which  P'  makes  with  the  resultant. 


4. — Required  the  angle  under  which  two  components  whose    inten- 
sities are  denoted  by  5   and  7  should   act,  to  give  a  resultant  whose 

intensity  is  represented  by  9. 

Ans.  t^°  U' 

'ik'  is' 3: 
5. — From    Equation    (56)    it    appears    that    the   resultant    of    two 

components  applied  to  the    same    point,  is    greatest   when  the    angle 

made  by  their  directions  is  0°,  and  least    when    180°.     Required   the 

angle   under    which   the    components   should    act,    in    order   that  the 

resultant    may    be    a   mean   proportional    between   these  values;   and 

also  the  angle  which  the  resultant  makes  with  the  greater  component. 

Call  P',  the  greater  component. 


Ans. 


-1       P" 

8  =  cos    -  — . 

.-1  P" 
9   =  sin    -^' 


6. — Given  a  force  whose  intensity  is  denoted  by  17.     Required   the 
two  components  which  make  with  it  angles  of  27°  and  43°.,,  ^'  ^^-'■^ 

§  88. — The  theorem  of  the  parallelogram  of  forces,  just  explained, 
enables  us  to  determine  by  an  easy  graphical  construction  the  in- 
tensity and  direction  of  the  resultant  of  several  forces  applied  to  the 
same  point. 

Let  P',  P'\  P'",  &c.,  be 
several  forces  applied  to  the 
same  point  m.  Upon  the 
directions  of  the  forces,  lay 
off  from  the  point  of  ap- 
plication distances  propor- 
tional to  the  intensities  of 
the  forces,  and  let  these  dis- 
tances represent  the  forces. 
From  the  extremity  P'  of 
the  line  mP\   which   repre- 


^ 


,iD,v 


T4  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

sents  the  first  force,  draw  the  line  P'  n  equal  and  parallel  to  m  F" 
which  represents  the  second,  then  will  the  line  joinmg  the  extremity 
of  this  line  and  the  point  of  application,  represent  the  resultant  of 
these  two  forces.  From  the  extremity  n,  draw  the  line  nn^  equal 
and  parallel  to  niF'"  which  represents  the  third  force;  mn'  will 
represent  the  resultant  of  the  first  three  forces.  The  construction 
bein<y  thus  continued  till  a  line  be  drawn  equal  and  parallel  to 
every  line  representing  a  force  of  the  system,  the  resultant  of  the 
whole  will  be  represented  by  the  line,  (in  this  instance  m  n"),  join- 
ing the  point  of  application  with  the  last  extremity  of  the  last 
line  drawn.  Should  the  line  which  is  drawn  equal  and  parallel  to 
that  which  represents  the  last  force,  terminate  in  the  point  of  appli- 
cation, the   resultant  will   be   equal   to   zero. 

The  reason  for  this  construction  is  too  obvious  to  need  expla- 
nation. 

§89. — If  the  forces  still  be  supposed  to  act  in  the  same  plane, 
but  upon  different  points  of  the  plane,  the  first  of  Equations  (49) 
takes   the   form, 

Yx  -  A"y  =  2  [P'  (cos  /3'  x'  -  cos  a'  y')  ], 
thus,  differing  from  Equation  (55),  in  giving  the  equation  of  the  line 
of  direction    of    the    resultant    an   independent     term,     and    showing 
that   this  line  no  longer  passes  through  the  origin.     It  may  be  con- 
structed from   the  above  equation. 

g90._To  find  the  resultant  in  this  case,  by  a  graphical  construc- 
tion, let  the  forces  P', 

P",  P'"    &c.,    be    ap-  ,  „ 

plied  to  the  points  m',       N  ,A       "7 

m",  m'",  &c.,   respec-  ^^  /    \  / 

tively.      Produce    the  /'    \,^  ^,/- ^ 

./?  /''     ) I /^ 

directions  of  the  forces        -^r  yO    \  /  \^ 

P'   and   P"   till    they  \^  \  /    _\  ^. 

meet   at   0,   and   take 

this   as   their  common 

point    of    application  ; 

lay  off  from    0,    on  the  lines   of  direction,  distances  0  S  and  0  S', 


MECUANICSOF     SOLIDS.  75 

proportional  to  the  intensities  of  the  forces  P'  and  P",  and  construct 
the  parallelogram  OS  US',  then  will  OR  represent  the  resultant  of 
these  forces.  The  direction  of  this  resultant  being  produced  till  it 
meet  the  direction  of  the  force  P"\  produced,  a  similar  construction 
will  give  the  resultant  df  the  first  resultant  and  the  force  P'", 
which  Avill  be  the  resultant  of  the  three  forces  P\  P"  and  P'" ; 
and   the  same  for   the   other  forces. 

OF   r,VJtATJ.EL   FOKCES. 

I  91. — If  the  forces  act  in   parallel    directions, 
a'  =    a."  =    a.'"  =  &c., 

/3'  =  /3"  =  13'"  =  &c., 
y'  =z   j"  =  y'"  =  &c., 

and  Equations  (41)  become, 

X  =  (P'  +  P"  +  P'"  +  &e.)  cos  a', 
Y  =  [P'  +  P"  +  P'"  +  &c.)  cos/3', 
Z  =  (P'  +  P"  +  P'"  +  &c.)  cos  7' ; 

these  values  in  Equation  (47)  give, 

i2  =  ±  1  .^J{P'  -\-  P"  +  P'"  +  &c.)2  (cos2  a'  +  cos2  /S'  +  cos^  7'), 

but, 

cos2  a'  +  cos2  ,S'  +  cos^  7'  =  1  ; 
hence, 

i2  =  P'  +  P"  +  P'"  +  &c.  ......     (60) 

If  some  of  the  forces  as  P",  P'",  act  in  directions  opposite  to 
the  others,  the  cosines  of  a"  and  a,'"  will  be  negative  while  they 
have   the   same   numerical  value ;  and   the   last  equation  will  become 

R  ^  P'  -  F'  -  P'"  +  &c. 

Whence  we  conclude,  that  the  resultant  of  a  number  of  parallel 
forces  is  equal  in  intensify  to  the  excess  of  the  sujn  of  the  inten- 
sities of  those  which  act  in  one  direction  over  the  sutn  of  the 
intensities   of  those   wKich   act   in   the   opposite  direction. 


76     ELEMENTS  OF  ANALYTICAL  MECHANICS. 

1 92._The  values  of  R,  X,  Y  and  Z  being  substituted  in  Equa- 
tions  (48)  give, 

_  (P^  +  P"  +  P'"  +  &c.)  cos  a'  ^  ^^^  ^, 

cos  a  —     p,  _^  p,,  j^  p,n  j^  ^^^  ' 

_  (P'  +  P"  +  P-  +  &0.)  COS  ^-  ^  ^„^^, 

/pr   ^   p.;   ^    pn,   ^    ^(.  >)  (.Qg       r 

rns  r  —  -^ ■ ■ ' —  =  COS  y  . 

COS  c  —     p,  _j_  p„  _^  p,,/  _^  ^^^  / 

The  denominator  of  these  expressions,  being  the  resultant,  is  essen- 
tially positive ;  the  signs  of  the  cosines  of  the  angles  a,  h  and  c, 
will,  therefore,  depend  upon  the  numerators ;  these  are  the  compo- 
nents  parallel  to   the  three  axes. 

Hence,  the  resultant  acts  in  the  direction  of  those  forces  whose 
cosines  are  negative  or  positive  according  as  the  sum  of  the  former 
or  latter  forces   is   the  greater. 

§93.— The  forces  being   still   parallel,  Equations  (42)  reduce  to, 
f       (P'  x'  +  P"  x"  +  P'"  x'"  +  &c.)  cos  /3' 

^.-C  cos  6  —  Py  cos  a  =  •<  /„,     ,     ,      r>,,     „     ,      nm     trr     I      s      \  r 

^  i  —  (P  y    4-  P'  y    +  P     y      +  &c.)  cos  a' 

(       {P'  z'  -f  P"  z"  +  P'"  z'"  +  &c.)  COS  a' 

Hz  cos  a  —  Marcos  C   =  ■<  ,  tm     ,     ,      nn     n     \      -dih     irr     ,      s      \  / 

I  —  [P'  X    +  P    X    +  P     X     4-  &c.)  cos  y 

f       {P'y'  +  P"y"  +  P'"/"  +  &c.)  cos  f 
i2  y  cos  c  -  P  .  cos  6  =  -j  _  ^^,  ^,  ^  P"  ."  +  P'"  z'"  +  &c.)  cos  ^^ 

but, 

cos  5  =  cos  /3', 

cos  a  =  cos  a', 

cos  C  =  cos  v'  ; 

Substituting  the  second  members  of  these  last  equations  for  the 
first  in  the  equations  immediately  preceding,  and  transposing  all  the 
terms   to  the  first  member,  we  obtain, 

[Bx  -  {P'x'  +  P"x"  -f  P"'x"'  +  &c.)]  cos/3'  I  _ 

-  [Py  -  {P'y'  +  P"y"  +  P'"  y'"  -f-  &c.)]  cos  aM  ~    ' 

[P  0  -  (P'  s'  +  P"  z"  +  P'"  2'"  +  &c.)]  cos  «' )  _  Q 

-  [Px  -  (P'  a;'  -{-  P"x"  +  P"'x"'  +  &c.)]  cos  7'  )  ' 

IRy  -  {P'y'  +  P"  y"  +  P'" y'"  +  &^c.)]  cos  7  )       ^ 
-{Rz  -  (P'  2'  +  P"  z"  +  P'"  £'"  +  &c.)]  cos/3'  f 


MECHANICS    OF    SOLIDS.  77 

These  equations  must  be  satisfied,  whatever  may  be  the  angles 
which  the  common  direction  of  the  forces  makes  with  the  co-ordinate 
axes,  and  this  can  only  be  done  by  making  the  co-efficients  of  the 
cos  a',  cos^''  and  cosy',  (either  two  of  the  latter  being  arbitrary), 
separately  equal   to   zero.     Hence, 

Rx  =  P'x'  4-  P"J^"  +  P"'x"'  +  &c. 

Rij  =  P'y'  +  P"y"  +  P"'y"'  +  &c.    j^    •     •     '     •   (61) 

Rz  =  P'z'  +  P"z"  +  P"'z"'  +  &c. 

The  forces  being  given,  the  value  of  R,  §91,  becomes  known, 
and  the  co-ordinates  ar,  y,  2,  are  determined  from  the  above  equations ; 
these  co-ordinates  will  obviously  remain  the  same  whatever  direction 
be  given  to  the  forces,  provided,  they  remain  parallel  and  retain  the 
same  intensity  and  points  of  application,  these  latter  elements  being 
the  only  ones  upon  which  the  values  of  ar,  y,  z,  depend. 

The  point  whose  co-ordinates  are  x,  ?/,  z,  which  is  the  point  of 
application  of  the  resultant,  is  called  the  centre  of  parallel  forces,  and 
may  be  defined  to  be,  that  point  in  a  system  of  parallel  forces  through 
which  the  resultant  of  the  system  tvill  always  pass,  %vhatever  be  the 
direction  of  the  forces,  2)^ovided,  their  intensities  and  points  of  appli- 
cation remain   the  same. 

I  94. — Dividing  each  of-  the  above  Equations  by  R,  we  shall  have 
P'x'  -f  P"x"  +  P"'x"'  +  (fee. 


y 


P'   +  P"  +  P'"  +  &c. 
P'y'  +  P"y"  +  P"'y"'  +  &c. 

P'  -f  P"  -f  P'"  +  &c. 

P'z'  4-  P"z"  +  P"'z"'  +  &c. 
P'  +  P"  +  P^"  +  &c. 


(62) 


Hence,  either  co-ordinate  of  the  centre  of  a  system  of  parallel  forces 
is  equal  to  the  algebraic  suin  of  the  products  which  result  from  multi- 
plying the  intensity  of  each  force  by  the  corresponding  co-ordinate  of  its 
point  of  apjylication,  divided  by  the  algebraic  sum  of  the  forces. 

If  the   points  of  application  of  the   forces   be   in   the   same   plane. 


78 


ELEMENTS    OF    ANALYTICAL    MECHANICS, 


the   co-ordinate  plane   ay,   may   be   taken  parallel   to   this   plane,   in 
which  case 


and, 


Z"  =r  Z'"   =  Z""  &C. 


_    {P'  +  P"  +  P'"  +  &c.)  z'   _    , 
^  ~       P'  +  P"  +  P'"  +  &c.       ""  ^ 


from  which  it  follows  that  the  centre  of  parallel  forces  is  also  in  this 
plane. 

If  the  points  of  application  be  upon  the  same  straight  line,  take 
the  axis  of  x  parallel  to  this  line ;  then  in  addition  to  the  above  results, 
we  have 

/  =  y"  =  y'"  =  &c. ; 
and, 

(P'  -f  P"  +  P'"  +  &e.)  y 


y  = 


y 


P'  +  P"  +  P'"  +  &c. 

whence,  the  centre  of  parallel  forces  is  also  upon  this  line. 

§  95. — If  we  suppose  the  parallel  forces  to  be  reduced  to  two,  viz. 
P'  and  P",  we  may  assume  the  axis  x  to  pass  through  their  points 
of  application,  and  the  plane  xy  to  contain  their  directions,  in  which 
case,  Equations  (60)  and  (61)  become, 

P  =  P'  +  P" 

Bx  =  P'x'  +  P"x" 
2  =  0    and   y  =  0. 

Multiplying  the  first  by  x',  and  subtracting 
the  product  from  the  second,  we  obtain 

E(x  -  x')  =  P"  {x"  -  .r')  .  .  (a) 

Multiplying  the  first  by  x"  and  sub- 
tracting the  second  from  the  product, 
we  get 


E  Or"  -  x)  =  P'  {x"  -  x')     . 


(M 


Denoting  by  ^S"  and  S",  the  distances  from  the  points  of  application 


MECHANICS    OF    SOLIDS. 


79 


of  P'  and  P"  to  that  of  the  resultant,  which  are  x  —  x'  and  x"  —  x 
respectively,  we  have 

x"  -  x'  =  5'  +  S"  ; 
and  from  Equations  («)  and  (i),  there  will  result 

P'  :  P"  .  R  ::  S"  :  S'  :  S"  +  S'      .     .     .     .    (63) 

If  the   forces   act   in   opposite   directioivs,  then,  on   the   supposition 
that  P'  is  the  greater,  will 

E  =  P'  -  P" 
Rx  =  P'x'  -  P"x" 
0  =  0,   y  =  0. 

and  by    a   process    plainly  indicated    by 
what  precedes, 


k?>^ 


P'  :  P"  :R::S":  S'  :  S"  -  *S".  .     (64). 

From  this  and_,  Proportion  (63),  it  is 
obvious  that  the  point  of  application  of 
the  resultant  is  always  nearer  that  of  the 
greater  component;    and    that    when    the 

components  act  in  the  same  direction,  the  distance  between  the  point 
of  application  of  the  smaller  component  and  that  of  the  resultant,  is 
less  than  the  distance  between  the  points  of  application  of  the  com- 
ponents, while  the  reverse  is  the  case  when  the  components  act  in 
opposite  directions.  In  the  first  case,  then,  the  resultant  is  between 
the  components,  and  in  the  second,  the  larger  component  is  always 
between  the  smaller  component  and  the  resultant. 

And  we  conclude,  generally,  ihai  the  resultant  of  two  forces  which 
solicit  two  points  of  a  right  line  in  parallel  directions,  is  equal  in  inten- 
sity to  the  sum  or  difference  of  the  intensities  of  the  components,  accord- 
ing as  they  act  in  the  same  or  opposite  directions,  that  it  always  acts 
in  the  direction  of  the  greater  component,  that  its  line  of  direction  is 
contained  in  the  plane  of  the  components,  and  that  the  intensity  of  either 
component  is  to  that  of  the  resultant,  as  the  distance  between  the  point 
of  application  of  the  other  component  and  that  of  the  resultant,  is  to 
the  distance  between  the  points  of  apjylication   of  the  components. 


80 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


§96. — Examples. — 1.    The   length   of    the    line    m' m"    joining   the 
points  of  application  of  two  jDarallel  forces 
acting  in  the  same  direction,  is  30  feet ;    the  X. 

forces  are  represented  by  the  numbers  15 
and  5.      Required  the  intensity  of  the  re- 
sultant, and  its  point  of  application. 
i2  =  P'  +  P"  =  15  +  5  =  20  ; 
B    :  P'  ::  m"  m'  :  m"  o, 


.P" 


J'. 


T/lT 


20  :  15  ::  30  :  m"  o  =  22,5  feet. 

A  single  force,  therefore,  whose  intensity  is  represented  by  20,  applied 
at  a  distance  from  the  point  of  application  of  the  smaller  force  equal 
to  22,5  feet,  will  produce  the  same  effect  as  the  given  forces  applied 
at  m"  and  m'. 

2. — Required  the  intensity  and  point 
of  application  of  the  resultant  of  two 
parallel  forces,  whose  intensities  are  de- 
noted by  the  numbers  11  and  3,  and 
which  solicit  the  extremities  of  a  right 
line  whose  length  is  16  feet  in  opposite 
directions. 


E  =  P'  -  F"  =  n  -  S  =  8, 
p,  _  p"  :  p'  :  :  m"  m'  :  m"  o 


P'  .  m"  m' 


=  22  feet. 


P'  -  P" 

3. — Given  the  length  of  a  line  whose  extremities  are  solicited  in 
the  same  direction  by  two  forces,  the  intensities  of  which  differ  by 
the  w'*  part  of  that  of  the  smaller.  Required  the  distance  of  the 
point  of  application  of  the  resultant  from  the  middle  of  the  line. 
Let  2  I,  denote  the  length  of  the  line.     Then,  by  the  conditions, 


P' 


B 


n  V      «.      / 

Ql  -\-   1\  p,,     .     pn  __ 

V     ?i     / 


J^ 


n 

'n  +  1 
n 

/2«_+J_\  p„  _  ^„      2^  .  „,/o  ^ 
V       ?J       / 

2«?  1 


%i  +  1 


Inl 


%i  +  1 


CO  =  Z  — 


%-^i  +1       2<i  +  1 


I. 


MECHANICS     OF    SOLIDS. 


81 


§97. — The  rule  at  the  close  of  §95,  enables  us  to  deterrnuie  by  a 
very  easy  graphical  construction,  the  position  and  point  of  application 
of  the  resultant  of  a  number  of  parallel  forces,  whose  directions, 
intensities,  and  points  of  application  are  given. 

Let  P,  P',  P",  F",  and  P'\ 
be  several  forces  applied  to  the 
material  points  m,  m',  m",  m"\ 
and  171",  in  parallel  directions. 
Join  the  points  m  and  m'  by  a 
straight  line,  and  divide  this  line 
at  the  point  o,  in  the  inverse 
ratio  of  the  intensities  of  the 
forces  F  and  F' ;  join  the  points 
o  and  m"  by  the  straight  line 
ow",  and   divide    this   line  at  o', 

in  the  inverse  ratio  of  the  sum  of  the  first  two  forces  and  the  force 
F"  ;  and  continue  this  construction  till  the  last  point  771"  is  included, 
then  will  the  last  point  of  division  be  the  point  of  application  of  the 
resultant,  through  which  its  direction  may  be  drawn  parallel  to  that 
of  the  forces.  The  intensity  of  the  resultant  will  be  equal  to  the 
algebraic  sum  of  the  intensities  of  the  forces. 

The  position  of  the  point  0  will  result  from  the  proportion 


F  -\-  F'  :  F'  : :  m  m'  :  m  0 
that  of  0'  from 

P  +  P'  +  P"   :   P"  :  :  0  m"  :  0  0'  = 
that  of  0"  from 

F  +  F'  +  F"  -  F"  :  -  F"  :  0'  m'"  :  0'  0" 
and  finally,  that  of  0'"  from 
p_(_P'4.p"_P"'-^pi-  :  P'^  ::  0"  m"  :  0"  0'"  = 


F  +  F' 


F+F  +  F' 


F"  .  0'  m' 


F+  F+  F"  —  F 


)>"'        r>"  /HI  '▼ 


P4_P'+P"_p'"  +  p» 


82  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


OF    COTTPLES. 

§98. — When  two  forces  P'  and  P"   act  in  opposite  directions,  the 
distance  of  the  point  o,  at  which  the  resultant 
is   applied,  from    the    point   m\    at   which   the 
component   P'   is    applied,  is   found   from   the 
formula 


,  m    m 

m  0  ■=. 


P'  -  P' 


and    if    the    components    P'    and   P"   become 

equal,  the    distance   m'  o   will   be   infinite,   and 

the  resultant,  zero.     In  other  words,  the  forces 

will    have    no    resultant,  and  their  joint  effect 

will  be  to  turn  the  line    m"  m\   about  some  point  between  the  points 

of  application. 

The  forces  in  this  case  act  in  opposite  directions,  are  equal,  but 
not  immediately  opposed.  To  such  forces  the  term  couple  is  applied. 
A  couple  having  no  single  resultant,  their  action  cannot  be  compared 
to  that  of  a  single  force. 

§99. — The  analytical  condition,  Equation  (46),  expressive  of  the 
existence  of  a  single  resultant  in  any  system  of  forces,  will  obviously 
be  fulfilled,  when 

X  -  0,     T  =  0,    and  Z  ^  0. 

But  this  may  arise  from  the  parallel  groups  of  forces  whose  sums 
are  denoted  by  X,  Y,  and  Z,  reducing  each  to  a  couple.  These  three 
couples  may  easily  be  reduced  by  composition  to  a  single  couple, 
beyond  which,  no  further  reduction  can  be  made.  It  is,  therefore,  a 
failins  case  of  the  general  analytical  condition  referred  to. 

WOEK   OF   THE   RESULTANT   AND   OF   ITS    COMPONENTS. 

§  100. — We  have  seen  that  when  the  resultant  of  several  forces 
is  introduced  as  an  additional  force  with  its  direction  reversed,  it 
will   hold   its   components   in    equilibrio.      Denoting   the   intensity   of 


MECHANICS    OF    SOLIDS.  83 

the   resultant    by   R,    and   the   projection   of    its   virtual   velocity  by 
5r,  we   have   from   Equation  (29), 

-  R5r  +  P.520  +  P'.op'  +  P".8p"  +  &c.  =  0, 
or, 

R8r  =  RSp  +  P'  Sp'  +  P"  Sjy  +  &c.,  ....     (G5) 

in   which  P,  P'  P'\  &c.  are  the   components,  and  ^  'p^  5 p'  5p'\  &c. 
the   projections  of  their  virtual  velocities. 

§  101. — Now,  the  displacement  by  which  Equation  (29)  was  de- 
duced, was  entirely  arbitrary  ;  it  may,  therefore,  be  made  to  conform 
in  all  respects  to  that  which  would  be  produced  by  the  components 
P,  P',  &c.,  acting  without  the  opposition  of  the  force  equal  and 
contrary  to  their  resultant;  and  writing  dr  for  S r,  dp  for  Sp,  &;c., 
Equation  (65)  will   become 

Rdr  =Pdp  +  P'dp'  +  P"dp"  +  &c.,  .     .     •     (GO) 
and   integrating, 

f]Rdr  :^  fPdp  +  J  P'dp'  +  fP"dp"  +  &c.,  .     .     (67) 

in  which  i2,  P,  P',  &c.  may  be  constant  or  functions  of  r,  ^9,  p',  &c., 
respectively. 

From  Equations  (66)  and  (67),  it  appears  that  the  quantity  of 
work  of  the  resultant  of  several  forces  is  equal  to  the  algebraic  sura 
of  the  quantities  of  work  of  its  components. 

Again,  replacing  P8p,  P'dp',  &c.  in  Equation  (65),  by  their  values 
in   Equation  (31),  and  writing  dr  for  (Jr,  dp  for  5 p,  &e.,  we  find, 

fRdr=if2P.cosa.dx+f^P.eos(3.di/+f:SP.cosy.dz,  ■  -  (6S) 

in  which  R   may  be   constant   or   a  function  of  r;  jP,  constant  or   a 
function   of  x,  y,  2,  &;c. 

If  the  forces  be   in   equilibrio,  then  will  R  =  0,  and, 

iP.eosa.dx  +  IIP.  COS  f3.dt/  -\-  Z P.  co^y.dz  =  0.  ■     •     (69) 


84 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


MOMENTS. 


S  102. If  the  forces  act  in  the  same  plane,  or  in  parallel  planes, 

the  axis  of  z  may  be  assumed  perpendicular  thereto,  in  which    case. 

cos7  =  cos/=:&c.=0;  cos6=smc(;  cos/3'=sina';  cos/3"=siua" &c., 

and   the  first  of  Equations  (42)  becomes, 

R{x.%ma-  y.  cos  a)  =  2  P' .  {x'  sin  a'  -  y'  cos  a')    •  •  (70) 

Denote  by  if,  the  length 
of  the  line  M A^  drawn  from 
the  point  of  application  M 
of  i2,  perpendicular  to  the 
axis  z,  and  by  (p,  the  angle 
which  this  line  makes  with 
the  axis  x.  Multiplying  and 
dividing  the  first  member  of 
the  above  equation  by  iT,  and 
reducing  by   the  relations, 


^-cos9;  ^ 


sni  9  ; 


and 


'  (sin  a  cos  (p  —  cos  a  sin  (pj  =  sin  ((p  —  a), 
there  will  result, 

—  It{x  sin  a  —  y  cos  a)  —  B.H.  sin  ((p  —  a). 

But  if  a  line  A  0,  be   drawn  from   the   point  A,  perpendicular  to 
the  line  of  direction  of  R,  and  its  length  be  denoted  by  K,  then  will 

H .  sin  {^  —  a)  ^:^  K\ 

■which,  in  the  above   equation,  gives 

— /2  (a; .  sin  a  —  y  cos  a)  =:  i2 . -ff". C^O 

In    the    same   way,   denoting    the    lengths   of    the    perpendiculars 
drawn  from   the   points  in   which  the   axis    z,   pierces   the   planes  of 


MECHANICS     OF    SOLIDS. 


85 


the    forces   P',    P",    &c.,    to    their    respective    lines   of   direction  by 
k\  k",  &c.,  -will, 

2  F'.  {x'  sill  a'  -  y'  cos  a')  =^P'.k';     -     -     •     (72) 
and   Equation  (65)  may  he  written, 

R.K  ^-ZP'.k'. (73) 

§  103.— The  lines  K,  k', 
(fee,  are  called  the  lever 
arms  of  the  forces  R,  P', 
&c.,  and  Equation  (73) 
shows  that  the  quantity 
of  work  of  the  resultant  of 
several  forces  acting  in 
parallel  planes,  and  through 
a  distance  equal  to  its  lever 
arm,  is  equal  to  the  alge- 
braic sum  of  the  quanti- 
ties of  work  of  its  comjjo- 
nents  acting    through  distances   equal   to    their  respective  lever  arms. 

§  104. — The  product  of  the  intensity  of  a  force  by  its  lever  arm, 
is  called   the  moment  of  the  force. 

A  line  perpendicular  to  the  plane  of  the  force  and  its  lever  arm, 
and  through  the  extremity  of  the  latter  most  remote  from  the  line 
of  direction  of  the  force,  is  called  the   moment   axis. 

The  point  in  which  the  moment  axis  pierces  the  plane  of  the 
force   and   lever   arm,  is   called   the   centre  of  moments. 

A  line  through  the  centre  of  moments  and  oblique  to  the  plane 
of  the  force   and   its   lever   arm,   is  called    a   component   axis. 

The  moment  of  the  resultant  of  several  forces,  is  called  the  result- 
ant moment. 

The  moments  of  the  several  components,  are  called  component 
moments— i\^Q  corresponding  axes  being  called  respectively  resultant 
and   component  axes. 

g  105. — If  the  moment  axis  be  fixed,  the  virtual  velocities  will 
be  arcs   of  circles. 


86      ELEMENTS  OF  ANALYTICAL  MECHANICS. 

Let  M  be  the  point  of  application 
of  the  force  R;  MR,  its  direction; 
u4,  its  centre  of  moments  ;  MN  its 
virtual  velocity;  MO,  the  projection 
of  the  latter,  and  AD  =  K,  the 
lever   arm. 

The    virtual    moment,   and    therefore    the   elementary   quantity    of 
work  of  R  will  be. 

R.MO. 

Denote  the    space   described   at   the   unit's    distance  by  ds,,  then 

will, 

MN=i  H.ds^, 

MO  =  H.ds^.  cos  NMO; 

but  because  AM   and  AD   are   respectively   perpendicular   to   MN 
and  MO,  the  angle  OMN  is  equal  to    the   angle  DAM,  and 

cos  NM  0  =  jj\ 

which  substituted  above,  gives 

M0=  K.ds,, 

and  multiplymg  by   R, 

R.MO  =  R.K.ds,. 

That  is  to  say,  the  elementary  quantity  of  work  -performed  by  a 
force  while  its  point  of  application  is  constrained  to  turn  about  its 
moment  axis,  is  equal  to  the  moment  of  the  force  multiplied  by  the 
differential  of  the   arc   described  at   the   xtnii's   distance  from   this  axis. 

§  106.— Multiplying   both  members   of   Equation   (73)  by  ds^,  we 

get, 

R.K.ds,  =  S.P'.k'.ds^, 

and  integrating, 

/R.K.ds,  =  f^P'.k'ds, (74) 


MECHANICS    OF    SOLIDS. 


87 


§  107. — Tlic  effect  of  a  force  acting  at  the  end  of  its  lever  arm, 
is  to  produce  rotation  about  the  other  end  as  the  centre  of  moments, 
supposed  fixed ;  the  resistance  at  the  centre  of  moments  is  equal 
and  contrary  to  the  action  of  the  force ;  the  action  of  the  force  and 
this  reaction  form,  therefore,  a  couple,  and  the  lever  arm  of  the  force 
is   also    called  the  lever  arm  of  the  couple. 

The  moments  of  the  forces  which  urge  a  body  to  turn  in  opposite 
directions   about   any   assumed   axis  must  have  contrary  signs. 

The  sign  of  P' p',  or  its  equal  P'  cos  a',  y'  —  P'  cos  /3',  x',  depends 
upon  the  angles  which  the  direction  of  the  force  makes  with  the 
axes  and  upon  the  signs  and  relative  values  of  the  co-ordinates  of 
the   point  of  api^lication. 

Let  the  angles  which  the  direction  of  any  force  makes  with  the 
co-ordinate  axes  be  estimated  from  the  positive  side  of  the  origin  ; 
then,  if  the  angles  which  this  direction  makes  with  both  axes 
be  acute,  and  the  point  of  application  lie  in  the  first  angle, 
P'  cos  aJ .  y'  and  P'  cos  ^'  x',  will  be  positive,  and  if  the  first  of 
these  products  exceed  the  second,  the  moment  will  be  positive  ;  but 
if  the   latter  be   the   greater,  the  moment  will    be   negative. 

In  the  first  case,  it  is  vir- 
tually assumed  that  the  direc- 
tion of  P  will  cut  the  axis  y, 
on  the  positive,  and  the  axis 
.r,  on  the  negative  side  of  the 
origin,  while  in  the  latter,  the 
reverse  will  be  true. 

§108. — The  forces  being  sup- 
posed to  act  in  any  directions 
M-hatever  upon  a  solid  body, 
each   force    may    be    replaced 

by  its  three  components,  parallel  respectively  to  the  rectangular  axes 
r,  y,  z.  The  components  parallel  to  the  axis  z,  can,  §  102,  have  no  in- 
fluence to  produce  rotation  about  that  line,  and  the  effect  of  all  the  forces 
in  this  respect,  will  be  the  same  as  that  of  their  components  parallel  to 
the  axes  x  and  y.      But   these    act   in   planes   at   right   angles   to  the 


88  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

axis  z  which  axis  being  taken  as  their  moment  axis,  the  effects  of 
these  components  may  be  computed  by  Equations  (70),  (73),  (74); 
and  by  reference  to  Equations  (42)  and  (72),  it  will  be  seen  that 
the  quantities  X,  M,  iV,  are  the  algebraic  sums  of  the  moments  of 
all  the  forces  in  reference  to  the  axes  2,  y,  and   a;,  respectively. 


KESULTAITT     MOMENT, 

g  109. — The  forces  being  supposed  to  act  in  any  directions  whatever, 
join  the  point  of  application  of  the  resultant  R  and  the  origin  by 
a  right  line,  and  denote  its  length  by  H.  Multiply  and  divide  each 
of  the  Equations  (44)  by  H^  and  reduce  by  the  relation, 

X 

y  y 

^=COS^, 

in   which   ^,   I  and   e,  denote    the    angles    which    the    line   H  makes 
with  the  axes  ar,  y  and  z,  respectively ;  then  will 


R.H .  (cos  b  .  cos  Z,  —  cos  a .  cos  |)  =  X, 
R .  H .  (cos  a  .  cos  £  —  cos  c  .  cos  ^)  =  3/, 
R .  H .  (cos  c  .  cos  I  —  cos  b  .  cos  s)  =  iV.  _, 


(75) 


Squaring  each  of  these  Equations  and  adding,  we  find 

cos2  5  ^  cos2  ^  _  2  cos  i .  cos  a  .  cos  ^ .  cos  |  +  cos^  a .  cos^  |  ') 
R^ .  H'^\  4-cos2  ^  _  cos2  £  —  2  cos  a .  COS  c  .  COS  s  .  cos  I  4-  cos2 c  .  cos2  I  } 
+cos2  c  _  cos2  g  —  2  COS  &  .  COS  c  .  COS  g .  cos  £  +  cos2  5  ^  cos2  £  J 
=  Z2  +  3/^  +  iV^2 (76) 


But 


cos^  a  +  cos^  b  +  cos^  c  =  1, (^7) 

cos2  I    +  cos2  I  +  C0S2  £  =  1, (78) 

cos  a .  COS  ^  4-  COS  5  .  cos  |  +  cos  c  .  cos  £  =  cos  9,  .  (79) 


MECHANICS     OF    SOLIDS.  89 

the  angle  (p,  being   that   made   by  the   line  H,  with  the  direction  of 
the  resultant. 

Collecting    the    co-efficients  of    cos^  a,    cos^  &,  cos^  c,  and  reducing 
by  the  following  relations,  deduced   from   Equations  (78)  ;   viz.  : 

cos^  £  +  cos-  ^  =  1  —  cos2  ^, 
cos^  ^  +  cos2  s  =  1  —  cos^  I, 

C0S2  g  _|.   cos2  ^   =    1    —  C0S2  j^ 

we  find, 

R^.H^.  [1  —  (cos  a . cos  ^  +  cos  6  .  cos  ^  +  cos  c .  coss)2]z=Z2+iP+iV2 ; 

from  Equation   (T9), 

1  —  (cos  a .  cos  ^  +  cos  6  .  cos  ^  +  cos  c .  cos  sf  =  \  —  cos^  9  =  sin^  9 ; 

which  reduces  the  above  to 

E^.II'^ .  sin2  (p  =  Z2  +  i/2  +  N^. 

But  H"^ .  sin^  9  is  the  square  of  the  perpendicular  drawn  from  the 
origin  to  the  direction  of  the  resultant;  it  is,  therefore,  the  square 
of  the  lever  arm  of  the  resultant  referred  to  the  origin  as  a  centre 
of  moments.  Denoting  this  lever  arm  by  K,  we  have  after  taking 
the   square   root, 

R.K=  y/  L'  +  li^  +  iV^2 (80) 

That  is  to  say,'  the  resultant  moment  of  any  system  of  forces  is  equal 
to  the  square  root  of  the  sum  of  the  squares  of  the  sums  of  the  com- 
ponent moments,  taken  in  reference  to  any  three  rectangular  axes  through 
the  point  assumed  as  the  centre  of  moments. 

§  110.— Dividing    the    first  of  Equations    (75),  by    Equation  (SO), 
we  find, 

H  (cos  h .  cos  ^  —  cos  a .  cos  |) L 


K  yX2  +  M''  +  iV2 

The  effect  of  a  force  is,  §77,  independent  of  the  position  of  its 
point  of  application,  provided  it  be  taken  on  the  line  of  direction. 
Let  the  point  of  application  of  J?,  be  taken  at  the  extremity  of  its 


90  ELEMENTs'oF    ANALYTICAL    MECHANICS. 

lever  arm,  then  will  11  coincide  with  and  be  equal  in  length  to  K\ 
^  and  I  will  become  the  angles  which  the  lever  arm  makes  with  the 
axes  X  and  y,  respectively,  and  the  well  known  relation  obtained 
from  the  formulas  for  the  transformation  of  co-ordinates  from  one 
set  of  rectangular  axes  to  another,  will  give 

cos  A  =  cos  b .  cos  ^  —  cos  a .  cos  |. 

in  which  A  is  the  angle  the  resultant  axis  makes  with  the  axis  z  ; 
whence, 

cos  A  =  ^ (81) 

In  the  same  way,  denoting  by  B*  and  C  the  angles  which  the 
moment  axis  of  H  makes  with  the  co-ordinate  axes  y  and,*  respec- 
tively, will, 

M 
cos  B  =      ,,  (82) 

-/Z2    -f    J/2    +   iV^2 

cos  C  =     ,__        ^      (83) 

-/X2    +    J/2   _|_    ^2 

whence  we  conclude  that,  the  cosine  of  the  angle  which  the  resultant 
axis  makes  with  any  assumed  line  is  equal  to  the  sum  of  the  moments 
of  the  forces  in  reference  to  this  line  taken  as  a  coinponent  axis 
divided   by  the   resultant   moment.  • 

§111. — Multiplying  Equation  (81)  by  Equation  (80),  there  will 
result, 

R.K.cosA=zL (84) 

which  shows  that  the  component  moment  of  any  system  of  forces  in 
reference  to  any  oblique  axis  is  equal  to  the  j>'>'oduct  of  the  resultant 
moment  of  the  system  into  the  cosine  of  the  angle  between  the  resultant 
and  component  axes. 

For  the  same  system  of  forces  and  the  .same  centre  of  moments, 
it  is  obvious  that  E  and  K  will  be  constant ;  whence,  Equation  (80), 
the     sum   of   the    squares  of  the   sums   of   the    moments    in    reference 


MECHANICS     OF     SOLIDS.  91 

to  amj  three  rectangular  axes  through  the  centre  of  moments,  taken 
as  component  axes  is  a  cons/ant  quantity.  Also,  since  the  axis  z 
may  have  an  infinite  number  of  positions  and  still  satisfy  the  con- 
dition of  making  equal  angles  with  the  resultant  axis,  we  see 
Equation  (84),  that  the  sum  of  the  moments  of  the  forces  in  reference 
to  all  component  axes  which  make  equal  angles  with  the  resultant 
axis  xvill   be    constant. 


112. — Denote    by    ^',  ^",  &"\    the    angles    which    any   'component 

pectively,  a 


axis   makes   with    the    co-ordinate    axes   *,  y    and   ^  respectively,  and 


(1/  ^v/y  V)  -  "^^  f  S'^cf  +^£^  -/    ^^  p  ^^  / 
{J  J     c>-Jf^'yt/  --  -  Cr'^f"  cc/-^  V  -/  -^-^  7^  ^^^^  ^--^^ 

-  -jf  ^  rf  .-        ^z^^^  'Ai^<x^  u^i^  „  -v:  ^;>^  -  ^  .  r  ^  -   ?a    r'^  —  £ 

TRANSLATION   OF  EQUA'nONS    (^)    AND   (i?). 

§  113. — Equations  (J)  and  (5)  may  now  be  translated.'  They  express 
the  conditions  of  equilibrium  of  a  system  of  forces  acting  in  various 
directions  and  upon  different  points  of  a  solid  body.  These  condi- 
tions  are   six  in  number ;  viz. : 


90  ELEMENTS'OF    ANALYTICAL    MECHANICS. 

lever  arm,  then  will  H  coincide  with  and  be  equal  in  length  to  K', 
Z,  and  I  will  become  the  angles  which  the  lever  arm  makes  with  the 
axes  X  and  y,  respectively,  and  the  well  known  relation  obtained 
from  the  formulas  for  the  transformation  of  co-ordinates  from  one 
set  of  rectangular  axes  to  another,  will  give 

cos  A  =  cos  b .  cos  ^  —  cos  a .  cos  |. 

in  which  A  is  the  angle  the  resultant  axis  makes  with  the  axis  z  ; 
whence, 


reference  to  any  oblique  axis  is  equal  to  the  product  of  the  resultant 
moment  of  the  system  into  the  cosine  of  the  angle  between  the  resultant 
and  component  axes. 

For  the  same  system  of  forces  and  the  .same  centre  of  moments, 
it  is  obvious  that  E  and  K  will  be  constant ;  whence.  Equation  (80), 
the     sum   of   the    squares  of  the   sums   of   the    moments    in    reference 


MECHANICS     OF     SOLIDS.  91 

to  any  three  rectanx/ular  axes  through  the  centre  of  moments,  taken, 
as  component  axes  is  a  constant  quantity.  Also,  since  the  axis  z 
may  have  an  infinite  number  of  positions  and  still  satisfy  the  con- 
dition of  making  equal  angles  with  the  resultant  axis,  we  see 
Equation  (84),  that  the  sura  of  the  moments  of  the  forces  in  reference 
to  all  component  axes  xohich  make  equal  anyles  with  the  resultant 
axis  will   be   constant. 

§112. — Denote  by  &',  6",  &'",  the  angles  which  any  'component 
axis  makes  with  the  co-ordinate  axes  »,  y  and  ^  respectively,  and 
by  S  the  angle  which  the  component  and  resultant  axes  make  with 
each   other,  then  will 

cos  (J  =  cos -4  .cos  6'  +•  cos  B .  cos  6"  -f-  cos  C.  cos  d"\ 

multiplying   both   members   by  Ji .  K,  we   have 

U.K. cos 5  —  E.E.cos  A . cos 6'  +  R.K. cos B cos &"  +  R.K. cos  C. cos &'", 

But,  Equation  (84), 

R .  K .  cos  ^  =  Z, 
R .  K.  cos  B  =  J/, 
R .  K.  cos  C  =  iV; 

which   substituted   above,  gives 

R.K.  cos  8  =  L.  cos  &'  +  M.  cos  &"  +  iV.  cos  &'"   ■  •  •  (85) 

That  is  to  say,  the  vioment  of  the  resultant  in  reference  to  any  com- 
ponent axis,  is  equal  to  the  sura  of  the  ijroducts  arising  from  multiplyiny 
the  sum  of  the  moments  in  reference  to  the  co-ordinate  axes,  by  the 
cosines  of  the  angles  which  the  direction  of  the  component  axis  makes 
with   these  co-ordinate  axes,  respectively. 

TRANSLATION    OF   EQtJA'nONS   {A)   AND   {B). 

§  113. — Equations  [A)  and  {B)  may  now  be  translated.'  They  express 
the  conditions  of  equilibrium  of  a  system  of  forces  acting  in  vflrious 
directions  and  upon  different  points  of  a  solid  body.  These  condi- 
tions  are   six  in  number ;  viz. : 


92      ELEMENTS  OF  ANALYTICAL  MECHANICS. 

1, The   cdcjehraic   sum  of  the   components  of  the  forces   in   each  of 

any    three   rectangular   directions   must  be  separately   equcd   to   zero; 

2. Tlie  algebraic  sum  of  the  moments  of  the  forces  taken  in  refer- 
ence to  each  of  three  rectangular  axes  drawn  through  any  assumed 
centre  of  moments,  must  be   separately   equal  to  zero. 

If  the  extraneous  forces  be  in  equilibrio,  the  terms  which  measure 
the  forces  of  inertia  will  disappear,  and  these  conditions  of  equilibrium 
will   be  expressed  by 


2  P .  cos  a  =  0, 
2  P  cos  ^  =  0, 
2  P.  cos  7  =  0  ;^ 


2  P .  {x'  cos  13  —  y'  cos  a)  =  0, 
2  P.  [z'.  cos  a  —  x'  cos  y)  =  0, 
2  P .  {y'  cos  y  —  z'  cos  /3)  =  0.  ^ 


w 


{By 


The  above  conditions,  which  relate  to  the  most  general  action 
of  a  system  of  forces,  are  qualified  by  restrictions  imposed  upon 
the   state   of  the  body. 

§114. — If  the  body  contam  o,  fixed  point,  the  origin  of  the  mova- 
ble co-ordinates,  in  Equation  (40),  may  be  taken  at  this  point ;  in 
which  case  we  shall  have, 

8x^  =.  0, 
5y,  =  0, 
Sz,  =  0; 

and  it  will  only  be  necessary  that  the  forces  satisfy  Equations 
(5),  these  being  the  co-efficients  of  the  indeterminate  quantities  that 
do  not  reduce  to  zero.  Hence,  in  the  case  of  a  fixed  point,  the 
sum  of  the  moments  of  the  forces,  taken  in  reference  to  each  of  three 
rectanaular  axes,  passing  through  the  point,  imist  separately  reduce  to 
zero. 

Should   the   system   contain    two  fixed  points,  one  of  the  axes,   as 


MECHANICS    OF     SOLIDS.  93 

that  of  .r,  may  be  assumed  to  coincide  with  the  line  joining  these 
points,  in  which  case,  there  will  result  in   Equation  (40), 

Sx^  =  0,  S!f>  =  0, 
S^j,  =  0,  H  =  0- 
Sz^  =  0, 

and  it  will  only  be  necessary  that  the  forces  satisfy  the  last  Equa- 
tion in  group  (B) ;  or  that  the  sum  of  the  moments  of  the  forces  in 
reference  to  the  line  joining  the  fixed  points,  reduce  to  zero. 

If  the  system  be  free  to  slide  along  this  line,  Sx^  will  not  reduce 
to  zero,  and  it  will  be  necessary  that  its  co-efficient,  in  Equation 
(40),  reduce  to  zero  ;  or  that  the  algebraic  sum  of  the  components  of 
the  given  forces  parallel  to  the  line  joining  the  fixed  2^oints,  also  reduce 
to  zero. 

If  three  points  of  the  system  be  constrained  to  remain  in  a 
fixed  plane,  one  of  the  co-ordinate  planes,  as  that  of  xy,  may  be 
assumed  parallel  to  this  plane;    in  which  case, 

5z,  =  0, 
5  TO-  =  0, 
H-0; 

and  the  forces  must  satisfy  the  first  and  second  of  Equations  {A), 
and  the  first  of  {B)\  that  is,  the  algebraic  sum  of  the  compionents 
of  the  given  forces  parallel  to  each  of  two  rectangular  axes  parallel  to 
the  given  ^j/awe,  must  separately  reduce  to  zero,  and  the  sum  of  the 
moments  in  reference  to  an  axis  perpendicular  to  this  2^liine  must  reduce 
to  zero. 

CENTRE    OF    GRAVITY. 

§115, — Gravity  is  the  name  given  to  that  force  which  urges  all 
bodies  towards  the  centre  of  the  earth.  This  force  acts  upon  every 
particle  of  matter.  Every  body  may,  therefore,  be  regarded  as 
subjected  to  the  action  of  a  system  of  forces  whose  number  is  equal 
to  the  number  of  its  particles,  and  whose  points  of  application  have, 
with  respect  to  any  system  of  axes,  the  same  co-ordinates  as  these 
particles. 


94  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

The  weight  of  a  body  is  the  resultant  of  this  system,  or  the 
resultant  of  all  the  forces  of  gravity  which  act  upon  it,  and  is  equal, 
in  intensity,  but  directly  opposed  to  the  force  which  is  just  sufficient 
to  support  the  body. 

The  direction  of  the  force  of  gravity  is  perpendicular  to  the 
earth's  surface.  Tlie  earth  is  an  oblate  spheroid,  of  small  eccentri- 
city, whose  mean  radius  is  nearly  four  thousand  miles ;  hence,  as  the 
directions  of  the  force  of  gravity  converge  towards  the  centre,  it  is 
obvious  that  these  directions,  when  they  appertain  to  particles  of 
the  same  body  of  ordinary  magnitude,  are  sensibly  parallel,  since 
the  linear  dimensions  of  such  bodies  may  be  neglected,  in  compari- 
son with  any  radius  of  curvature  of  the  earth. 

The  centre  of  such  a  system  of  forces  is  determined  by  Equa- 
tions (G2),  §94,   which  are 

P'x'  +  P"x"  +  P"'x"'  +  &c. 


y, 


P'  + 

P" 

+  P"' 

+  &c. 

p 

y'  +  P 

"y" 

+  P'" 

y'" 

+  &c. 

P'  + 

p" 

+  P'" 

+ 

&c. 

p 

z'   +  P 

"z" 

+  P'" 

z'" 

+  &c. 

P'  +  p"  -I-  p">  4-  &c. 


(86) 


in  which  x^  y,  2^,  are  the  co-ordinates  of  the  centre ;  P',  P",  &c., 
the  forces  arising  from  the  action  of  the  force  of  gravity,  that  is, 
the  weights  of  the  elementary  masses  in\  m",  &c.,  of  which  the 
co-ordinates  are  respectively  x'  y'  z',   x"  y"  z",  &c. 

This  centre  is  called  the  centre  of  gravity.  From  the  values  of 
its  co-ordinates,  Equations  (8G),  it  is  apparent  that  the  position  of 
this  point  is  independent  of  the  direction  of  the  force  of  gravity  in 
reference  to  any  assumed  line  of  the  body;  and  the  centre  of  gravity 
of  a  body  may  be  defined  to  be  tliat  j^oint  throvgh  which  its  weight 
ahoays  passes  in  ivhatever  way  the  body  may  be  turned  in  regard  to 
the  direction  of  the  force  of  gravity. 

The  values  of  P\  P",  &:c.,  being  regarded  as  the  weights  w',  w", 
&c.,  of  the  elementary  masses  7n',  m",  &c.,  we  have.  Equation  (1), 

P'  =  w'  =  m'g'  ■    P"  =  to"  =  m"  g"  ;    P'"  =  w'"  =  m'"  g'"  ;   &c., 


MECIIAXICS    OF    SOLIDS. 

and,  Equations  (86), 

m'  g'  x'  +  m"  (j"  x"  +  m'"  g'"  x'"  +  &c. 


95 


y, 


m' 

'J' 

+ 

m" 

9" 

+ 

VI'"  g'" 

+  &c. 

m 

9'y' 

+ 

m 

"9' 

'v" 

+ 

m"'g" 

'y'".+  &c. 

m' 

9' 

+ 

m" 

9" 

+ 

m"'g"' 

+  &c. 

m' 

g'z' 

+ 

m 

V 

'  z" 

+ 

vi'"  g" 

'  z'"  4-  &:c. 

m'(7'  +  m"  g"  +  m'"^'"  +  &c. 


(87) 


§11G. — It  will  be  shown  by  a  process  to  be  given  in  the  proper 
place,  that  the  intensity  of  the  force  of  gravity  varies  inversely  as 
the  square  of  the  distance  from  the  centre  of  the  earth.  The  distance 
from  the  surface  to  the  centre  of  the  earth  is  nearly  four  thousand 
miles  ;  a  change  of  half  a  mile  in  the  distance  at  the  surfoce  would, 
therefore,  only  cause  a  change  of  one  four-thousandth  part  of  its 
entire  amount  in  the  force  of  gravity ;  and  hence,  within  the  limits 
of  bodies  whose  centres  of  gravity  it  may  be  desirable  in  practice  to 
determine,  the  change  would  be  inappreciable.  Assuming,  then,  the 
force  of  gravity  at  the  same  place  as  constant,  Equations  (87), 
become 

m'  x'  +  m"  x"  +  m'"  x'"  +  &c.    >, 


y> 


m'  + 

m" 

+ 

m'" 

+  &c. 

m' 

y'  +  m 

"y"  + 

m" 

y'"  +  &c. 

m'  + 

m" 

+ 

m'" 

+  &c. 

m' 

z'  +  '>^i 

'z" 

+ 

m'" 

z'"  +  &c. 

(88) 


'  vi'  +  vt,"  +  m'"  -\-  &c. 

from  which  it  appears,  that  when  the  action  of  the  force  of  gravity 
is  constant  throughout  any  collection  of  particles,  the  position  of  the 
centre  of  gravity  is  independent  of  the  intensity  of  the  force. 

§  117. — Substituting  the  value  of  the  masses,  given  in  Equation  (1)', 
there  will  result, 

v'  d'x'  +  v"d"  x"  +  v"'d"'x"'  +  &c.   ^ 


y> 


v'  d'  +  v"  d"  +  v'"  d'"  -f  &c. 

_  v'd'y'  +  v"  d"y"  +  v'"  d'"  y'"  +  d:c. 

^  v'd'  +  v"  d"  +  v"'d"'  -f  &c.         ' 

_  v'd'z'  +  v"  d"z"  +  v'"  d"'z"'  -\-  &c. 

~  v'  d'  +  v"  d"  +  v'"  d'"  +  &c.        ' 


(89) 


96 


ELEMENTS     OF    ANALYTICAI,    MECHANICS, 


and  if  the   elements  be  of  homogenous   density  throughout,  we   shall 
have, 

d'  =  d"  =  d'"  =  &c. ; 
and  Equations  (89)  become, 

v'x'  +  v"x"  +  v"'x'"  +  &c.   ^ 


y/ 


v'  + 

v" 

+ 

v'" 

+ 

&c. 

v' 

y' 

+  v"y"  + 

v" 

y'"  + 

&iC. 

v'  + 

v" 

+ 

v'" 

-L 

&c. 

V 

z' 

+  v"  z"  + 

v" 

z'"  +  &c. 

v'  +  v"  +  v'"  +  &c. 


(90) 


whence  it  follows,  that  in  all  homogeneous  bodies,  the  position  of 
the  centre  of  gravity  is  independent  of  the  density,  provided  the 
intensity  of  gravity  is  the  same  throughout. 

§  118. — Employing  the  character  2,  in  its  usual  signification,  Equa- 
tions (90),  may  be  written, 

^  {vx)    " 


y,  = 


2(.) 
2  {vy) 


T^»     > 


2(.) 

2  (v  z)  . 


(91) 


'  ~     2  (.)      J 
and  if  the  system  bo  so  united  as  to  be  continuous, 

X-   x.dV 


V 


y, 


pv' 

Jv"  y- 


dV 


pv' 
Jv''    ^ 


dV 


(92) 


§119. — If  the  collection  be  divided  symmetrically   by   the  plane 

xy,  then  will 

2(vz)  =  0, 


MECHANICS     OF    SOLIDS. 


97 


and,  therefore, 

^.  =  0  ; 
hence,  the  centre  of  gravity  -will  lie  in  this  plane. 

If,  at  the  same  time,  the  collection  of  elements  be  symmetrically 
divided  by  the  plane  x  z,  we  shall  have, 

2  (vy)  =  0, 

the  collection  of  elements  will  be  symmetrically  disposed  about  the 
axis  X,  and  the  centre  of  gravity  will  be  on  that  line. 

Although  it  is  always  true,  that  the  centre  of  gravity  will  lie  in 
a  plane  or  line  that  divides  a  homogeneous  collection  of  particles 
symmetrically ;  yet,  the  reverse,  it  is  obvious,  is  not  always  true, 
viz. :  that  the  collection  will  be  symmetrically  divided  by  a  plane  or 
line  that  may  contain  the  centre  of  gravity. 

Equations  (92)  are  employed  to  determine  the  centres  of  gravity 
of  all  geometrical  figures. 

THE   CENTRE   OF   GEAVrTY    OF   LINES. 

§  120. — Let  s  represent  the  entire  length  of  the  arc  of  any  curve, 
whose  centre  of  gravity  is  to  be  found,  and  of  which  the  co-ordi- 
nates  of  the   extremities   are   x\  y',  z',  and  x",  ?/",  z". 

To  be  applicable  to  this  general  cgsc  of  a  curve,  included  within 
the   given   limits.  Equations  (92)  become 


I.   xdx.^l+^  + 


d  x'~ 


dx- 


/■^'    ,     r     dV^     d~^ 

.V 

/,,  -''-V'  +  ^  +  rf^ 


(9J 


98  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

in  which 


=  /."  ^'y 


dip'         dz^ 

1  -\ 1 

dx^         dx^ 


(94) 


Example  1. — Find   the  j^osition  of  the  centre   of  gravity   of  a   right 
line.     Let, 

y  =z  a.  X  -\-  ^,,  Z 

z  =  a'x  -{-  13', 

be  the  equations  of  the 
line. 

Differentiating,  substi- 
tuting in  Equations  (94) 
and  (93),  integrating  be- 
tween the  proper  limits, 
and  reducing,  there  will 
result, 

x'  +  x' 


2 

a 

(•^' 

+ 

X") 

2 

a' 

.{X 

+ 

-") 

+  13', 


which  are  the  co-ordinates  of  the  middle  point  of  the  line ;  x'  y'  z' 
and  x"  y"  z",  being  those  of  its  extremities ;  whence  we  conclude 
that  the  centre  of  gravity  of  a  straight  line  is  at  its  middle  point. 

Example  2. — Find  the  centre  of  gravity  of  the  j^erimeter  of  a  polygon. 

This  may  be  done,  according  to  Equations  (90),  by  taking  the  sum 
of  the  products  which  result  from  multiplying  the  length  of  each  side 
by  the  co-ordinate  of  its  middle  point,  and  dividing  this  sum  by  the 
length  of  the  perimeter  of  the  polygon.  Or  by  construction,  as  fol- 
lows : 

The  weights  of  the  several  sides  of  the  polygon  constitute  a  system 
of  parallel  forces,  whose  points  of  application  are  the  centres  of 
gravity  of  the  sides.  The  sides  being  of  homogeneous  density,  their 
weights  are  proportional  to  their  lengths.     Hence,  to  fmd   the   centre 


MECHAXICS    OF     SOLIDS. 


99 


of  gravity  of  the  entire  polygon,  join  the  middle  points  of  any  two 
of  the  sides  by  a  right  line,  and  divide  this  line  in  the  inverse  ratio 
of  the  lengths  of  the  adjacent  sides,  the  point  of  division  will,  §  97, 
be  the  centre  of  gravity  of  these  two  sides;  next,  join  this  point 
with  the  middle  of  a  third  side  by  a  straight  line,  and  divide  this 
line  in  the  inverse  ratio  of  the  sum  of  first  two  sides,  and  this  third 
side,  the  point  of  division  will  be  the  centre  of  gravity  of  the  three 
sides.  Continue  this  process  till  all  the  sides  be  taken,  and  the  last 
point  of  division  will  be   the  centre  of  gravity  of  the   polygon. 

Find  the  position  of  the  centre  of  gravity  of  a  plane  curve. 
Assume  the  plane  of  x  y  to  coincide  with  the  plane  of  the  curve, 
in  which  case, 

d  z 


d  X 


=  0, 


and  Equations  (93)  and  (94)  become. 


y:::-V'''^^^ 


1  + 


rf^2 


y/  = 


Cy'^\J^  +  % 


(95) 


(90) 


Example  3. — Find  the   centre  of  gravity  of  a  circular  arc. 
Take    the    origin    at    the   centre    of  curvature,    and    the   axis    of  y 
passing   through    the   middle    point  of  the    arc.     The    equation  of  the 
curve   is. 


whence. 


y^  =  a"  —  x-^ 


dy 
dx 


wdiich  substituted  in  Equations  (95), 


100         ELEMENTS     OF    ANALYTICAL    MECHANICS. 

will  give  on  reduction, 

X,  =0, 

a  ix'  +  x")  . 

y        —      ^ , 

and  denoting   the   chord  of  the   arc  by  c  =  x'  +  x", 


x,  =  0, 
ac 


■whence  we  conclude  that  the  centre  of  gravity  of  a  circular  arc  is 
cm  a  line  drawn  through  the  centre  of  curvature  and  its  middle  point, 
and  at  a  distance  from  the  centre  equal  to  a  fourth  proportional  to 
the   arc,  radius  and  chord. 

Example  4. — Find  the  centre  of  gravity  of  the  arc  of  a  cycloid. 

The   radius  of  the  generating  circle  being   a,  the  differential  equa- 
tion of  the  curve  is, 


dx  =  -^ 


y  -dy 


^J2ay  —  y^ 


(«) 


the   origin  being  at  A,  and 

AB  being  the  axis  of  x.  ji 

Transfer  the  origin  to  C, 
and  denote  by  x'  y'  the  new 

co-ordinates,  the  former  being  estimated  in  the  direction  CD,  and  the 
latter  in  the  direction  D  B.     Then  will 


y  =  2a  -  x', 

X  =  at(  —  y' ; 

and  therefore, 

dx 

dy  ~ 

dy'               2a  —  x' 

dx'         ■y/2ax'  —  x'~ 

(«)' 


MECHANICS    OF    SOLIDS.  101 

this,  in    Equations  (9G)  and  (95),  gives,  omitting    the   accent   on   the 
variables, 

s    =    I  ,,  dx  \/  — » 

Jx  V        X 

r"'     1     /^ 

/   ,,     XdXK       — 
J  X  \         X 

/2a 

X 


y,  = 


Integrating  the  first  two  equations  between  the  limits  indicated, 
and  substituting  the  value  of  5,  deduced  from  the  first,  in  the  second, 
we  have, 

_  1  vv^3_-^vv3. 

""'   ""   '^'  -^X"     --y/x'     '' 

and  from   the  third  equation  we  have,  after    integrating  by  parts, 


substituting   the  value  of  rfy,   obtained   from   Equation  (a)',   and   re- 
ducing, there  will  result, 

sy^  =  2  s/'i-a  {y  ^ x  —  f  -y/'la  —  x.  dx), 

and  taking  the  integral  between  the  indicated  limits, 

sy,=2  V^%{V^'  -  /^)  +  !  (2a  -  x")!  -  |(2«  -  x')l]  ; 
hence,  replacing  s  by  its  value,  and  dividing, 

'{'/^    {2a-x"f  -{2a-x')'^ 

y.  =  %+  §  •  ^ — 


X     —  y  X 

Supposing  the  arc  to  begin  at  C,  we  have, 

x'  =  0, 
and, 


1  ~" 

3  "^    > 


/=y  +  :r7=^-[(-«-^")'  -2a-/2^], 


102  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

If  the  entire  semi-arc  from  C  to  ^  be  taken,  these  values  become, 

a:,  =  f  «, 

Taking  the  entire  arc  A  C  B,  the  curve  will  be  symmetrical  with  res- 
pect to  the  axis  of  x\  and  therefore, 

hence,  the  centre  of  gravity  of  the  arc  of  the  cycloid,  generated  hy  one 
entire  revohition  of  the  generating  circle,  is  on  the  line  which  divides 
the  curve  symmetrically,  and  at  a  distance  from  the  summit  of  the  curve 
equal  to  M^-thirds  of  its  height. 


'      THE    CENTKE    OF    GKAVITY    OF    SUKFACES. 

8  121. — Let  L  =  0,  he  the  equation  of  any  surface ;  L  being  a 
function  of  xyz;  then  will  dxdy,  be  the  projection  of  an  element 
of  this  surface,  whose  co-ordinates  are  xyz,  upon  the  plane  xy,  and 
if  6"  denote  the  angle  which  a  plane  tangent  to  the  surface  at  the 
same  point  makes  with  the  plane  xy,  the  value  of  the  element  itself 
will  be 


dx .  dy 

cos  6"  ' 

But  the  angle  which  a  plane 
makes  with  the  co-ordinate 
plane  xy,  is  equal  to  the 
angle  which  the  normal  to 
the  plane  makes  with  the 
axis  z,  and,  therefore, 


cos  6"  =:    ±1 


dL 

dz 


Vv^ 


dL^  ,  /'^V  _L  (UlS' 

dJJ    "^  \dv/    "^  Vrf2/ 


=  ±i-  '    '    (97) 


^i^^  '=     ,,r7-— ; 


\''/->C^^rl^ 


I    l-h: 


a^vf; 


MECHANICS     OF    SOLIDS. 

and  hence,  in  Equations  (92),  omitting  the  double  sign, 

« 
d  V  =:  d  X  •  d  y  •  to,    .     .     ,     . 

and  those  Equations  become, 


«>  = 


l/«      J  X 


Vi  = 


J  M      J  X 


iV'     />^ 


j,"Jx"    W. 


w . X .d  X . dy 


10  .  y  dx  .  dy 


z  .  dx  .  dy 


in  which, 


s  =  V  =       n      n  iv.dx.dy; 

J   V      V    X 


w  being  a  function  of  x,  y,  z. 

If  the  surface  be  plane,  the 
plane  oi  xy  may  be  taken  in  the 
surflice,  in  which  case, 

?y  =  1, 
.  =  0, 

and  Equations  (99),  and  (100),   be- 
come. 


V, 


J y"  J x"  dy  .xdx 
s 

f  f" 

J y"  J x"  dx  .y  dy 


s  =  f/'f^"  f^-^-^^y'    • 


103 


(98) 


(99) 


(100) 


A      J"  ^  J" 


ir—X 


(101) 


(102) 


which    the   integral    is    to    be    taken    first    with   respect    to   y,  and 


104 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


between  the  limits  y"  =  P  m"  and  y'  =  P  m' ;   then  in  respect  to  a;, 
between  the  limits  x"  =  AP'\  Snd  x'  =  AP'.    Hence 


/ 


2//  = 


."{y"  -y').xclx 


^L"{y"^-y")d: 


(103) 


s=fUy"-y')di 


(104) 


y'  and  y",  denoting  running  co-ordinates,  which  may  be  either  roots 
of  the  same  equation,  resulting  from  the  same  value  of  x,  or  they 
may  belong  to  two  distinct  functions  of  x,  the  value  of  x  being  the 
same  in  each.      For  instance,  if 


F    {xy)  =  0, 

be  the  equation  of  the  curve  n'  m"  n"  m\  it  is  obvious  that  between 
the  limits  x"  =  A  P"  and  x'  —  A  P\  every  value  of  .r,  as  A  P, 
must  give  two  values  for  y,  viz.:  y"  =  Pm"  and  y'  —  Pm'.      Or  if 

F{xy)  =  (i, 
F'  {xy)  =  0, 


be  the  equations  of  two  distinct 
curves  m"  n"  and  m'  n',  referred 
to  the  same  origin  A,  then  will 
y"  and  y'  result  from  these 
functions  separately,  when  the 
same  value  is  given  to  a;  in 
each. 


Example  1. — Required  the  position  of  the   centre   of  gravity   of  the 
area  of  a  triangle. 


MECIIAXiCS     OF    SOLIDS. 


105 


Let  ABC,  be  the  triangle. 
Assume  the  origin  of  co-ordi- 
nates at  one  of  the  angles  A, 
and  draw  the  axis  y  parallel  to 
the  opposite  side  B  C.  Denote 
the  distance  A  P  by  x\  and 
suppose, 

y"  =  ax, 
y'  =  ^^, 

to    be   the   equations   of  the    sides   A  C  and   A  B,   respectively,   then 

will 

y"    -y'  =  {a    -    h)  X, 

y"2  _  y'2  =  (a2  _  P)  x2, 

and, 


f    (a  —  b)  x^dx 
r  [a  —  h)  X  d  X 


y,  = 


^jy-y^)x^dx  ^  ^  (^^,),. 

/       [a  —  h)  X  dx 

J  xi 


whence  we  conclude,  that  the-  centre  of  gravity  of  a  triangle  is  on  a 
line  drawn  from  any  one  of  the  angles  to  the  middle  of  the  opjmsite 
side,  and  at  a  distance  from  this  angle  equal  to  two-thirds  of  the  line 
thus  drawn. 

'Exajnj)le  2. — Find  the  centre  of  gravity  of  the  area  of  any  polygon. 

From  any  one  of  the  angles 
as  A,  of  the  polygon,  draw  lines 
to  all  the  other  angles  except 
those  which  are  adjacent  on  either 
side;  the  polygon  will  thus  be 
divided  into  triangles.  Find  by 
the  rule  just  given,  the  centre  of 
gravity  of  each  of  the  triangles ; 


106  ELEliiENTS     OF    ANALYTICAL    MECHANICS. 

join  any  two  of  these  centres  by  a  right  line,  and  divide  this  line  in 
the  inverse  ratio  of  the  areas  of  the  triangles  to  which  these  centres 
belontr ;  the  point  of  division  will  be  the  centre  of  gravity  of  these 
two  triano-les.  Join,  by  a  straight  line,  this  centre  with  the  centre  of 
gravity  of  a  third  triangle,  and  divide  this  line  in  the  inverse  ratio 
of  the  sum  of  the  areas  of  the  first  two  triangles  and  of  the  third,  this 
point  of  division  will  be  the  centre  of  gravity  of  the  three  triangles.. 
Continue  this  process  till  all  the  triangles  be  embraced  by  it,  and  the 
last  point  of  division  will  be  the  centre  of  gravity  of  the  polygon; 
the  reasons  for  the  rule  being  the  same  as  those  given  for  the  deter- 
mination of  the  centre  of  gravity  of  the  perimeter  of  a  polygon,  it 
being  only  necessary  to  substitute  the  areas  of  the  triangles  for  the 
lengths  of  the  sides. 

Uxamjjle  3. — Determine    the   position    of  the   centre  of  gravity  of  a 
circular  sector.  ' 

The  centre  of  gravity  of  the  sec- 
tor will  be  on  the  radius  drawn  to 
the  middle  point  of  the  arc,  since  this 
radius  divides  the  sector  symmetri- 
cally. Conceive  the  sector  CAB,  to 
be  divided  into  an  indefinite  number 
of  elementary  sectors ;  each  one  of 
these  may  be  regarded  as  a  triangle 
whose  centre  of  gravity  is  at  a  dis- 
tance   from    the    centre    C,    equal    to 

two-thirds  of  the  radius.  If,  therefore,  from  this  centre  an  arc  be 
described  with  a  radius  equal  to  two-thirds  the  radius  of  the  sector, 
this  arc  will  be  the  locus  of  the  centres  of  gravity  of  all  the 
elementary  sectors ;  and  for  reasons  already  explained,  the  centre  of 
gravity  of  the  entire  sector  Avili  be  the  same  as  that  of  the  portion 
of  this  arc  which  is  included  between  the  extreme  radii  of  the  sector. 
Hence,  calling  r  the  radius  of  the  sector,  a  and  c  its  arc  and  chord 
respectively,  and  x^  the  distance  of  the  centre  of  gravity  from  the 
centre   C,  we  have, 

_fr.fc_2     r  .c 
'  4a  3       a 


MECHANICS     OF    SOLIDS. 


lOT 


The  centre  of  gravity  of  a  circular  sector  is  therefore  on  the  radius 
drcncn  to  the  middle  2)oi}it  of  the  arc  of  the  sector,  and  at  a  distance 
from  the  centre  of  curvature  equal  to  two-thirds  of  a  fourth  propor- 
tional   to   the   arc,  chord'  and  radius  of  the   sector. 

Exam2^h  4. — Find  the  centre  of  gravitij  of  a  circular  segment. 

Assume  the  origin  at  the  centre  C, 
and  take  the  axis  x  passing  through  the 
middle  point  of  the  arc,  the  centre  of 
gravity  in  question  will  be  on  this  axis, 
and,   therefore, 

y.  =  0. 

Let  A  B  HA  be   the  sector,  and 
y  =   ±  -y/  a"^  —  ar^, 

the  equation  of  the  circle,  the  origin  being 
at   the  centre  C,  then  will 


y'  =  —  -v/"'^  —  -^'^ 

and,  Equations  (103)  and  (104), 


or'  r-. 

J  a   V  «"^  —  x"^  .X  .d:. 


-f(a2-.r'^)2 


s  =  "^  /      -y/a^  —  X-  .dx.  =  —      u~  I-  —  sin       — )  —  x'  -yja^  —  x''^    , 

s   being   the    area  of   the    entire    segment.      Denoting   the   chord  A  B 
by  f,  we  have, 

whence, 


■y/d^  —  x^  =  I 


'  ~  12. s' 

and  we  conclude,  that  the  centre  of  gravity  of  a  circular  segment 
is  on  the  radius  drawn  to  the  middle  of  the  arc,  and  at  a  distance 
from  the  centre  equal  to  the  cube  of  (he  chord,  divided  by  twelve 
times    the    area    of  the    segment. 


108 


ELEMENTS  OF  ANALYTICAL  MECHANICS, 


Eeplacing   the   value   of  s,  and   supposing    x'    to   be   zero,  ia  which 
case   the   segment  becomes   a   semicircle,  we   shall  find, 

c  =  2  «, 
_  4a 

e  122. If  the  surface   be   one  of  revolution,  about   the    axis  x  for 

instance    it  will  be   symmetrical  with  respect  to  this  axis;  hence, 

and  if  Fixrj)  =  0,  be  the  equation  of  a  meridian  section  in  the 
plane  xy,  then  will  the  area  of  an  elementary  zone  comprised  be- 
tween  two   planes   perpendicular   to   the   axis  of  revolution   be, 

and  therefore,  Equations  (103)  and  (104), 


a;,  =  2* 


rJ'y^sF^'^'' 


(105) 
(lOG) 


Example  1. — Find 
the  position  of  the 
centre  of  gravity  of 
a  right  conical  sur- 
face. 

The  equation  of 
the  element  in  the 
plane  xy,  is,  assum- 
ing the  origin  at  the 
vertex. 


hence, 


y  =  ax; 
'  I  ,,  ax"^  dx  yl+o^ 

J  X 


2  -r  I  ,,  ax   dx  -y/  \  -\-  a- 


—  =  —  X- 
3 


MECPIAXICS    OF     SOLIDS. 


109 


Exaiiqile  2. — Required  the  posi- 
tion of  the  centre  of  yravity  of 
the   segment   of  a  sjjhere. 

Assuming  the  origin  at  the 
centre,  the  equation  of  the  me- 
ridian  curve  is, 

,,2    _   .,2   _  3-2  . 


whence, 


y^  =  w^  —  x^ 


y  dy  =  —  X  d  X, 

dy"^  x^ 

d  x^  ?/2' 


and, 


x' 

y  ,,  ax  d x 

x' 

/  ,,  ad  X 


."2    -r'2 


x"  4-  x' 


2  {x"  -  x')  2 


Hence,  the  centre  of  gravity  of  a  spherical  zone^  is  at  the  middle 
point  of  a  line  joining  the  centres  of  its  circular  bases.  And  in  the 
case  of  a  segment   it  is   only  necessary  to   make   x'  =  a,  ivkich  gives, 

_  x"  -}-  a 
^/  -        2      • 

So  that  the  centre  of  gravity  of  a  spherical  segment  is  at  the  middle 
of  the   ver-sine  of  its  surface. 


THE  CENTRES  OF  GRAVITY  OF  VOLUMES. 

§  123. — When  it  is  the  question  to  determine  the  centre  of  gravity 
of  the  volume  of  any  body,  we   have 

dV  =:  d X .dy .dz, 

and  Equations  (92)  become, 


px'     py'     nz' 

/  //   I  tr    I  ,,x.dy.dz.dx 

J  X       J  V        J  z 


110  ELEMENTS     OF    ANALYTICAL    MECHAJflCS. 

/x'     py'     pz' 
f,       ft       ,,y.dy.dz.dx 
U  y       V  z 


y.  = 


and. 


ytx<      py'     pz' , 
,,  /  ,r   I  ,,  z.dy .dz.dx 
X      J  V      J  z 

' V ' 

px'     py'     pz' 

n       n       I,  dy .dz.dx. 

u  X     J  y     J  z 


In  which  the  triple  integral  must  be  extended  to  include  the 
entire  space  embraced  by  the  surface  of  the  body ;  this  surface 
being   given   by  its  equation. 

If  the  volume  be  symmetrical  with  respect  to  any  line,  this  line 
may  be  assumed  as  one  of  the  co-ordinate  axes,  as  that  of  x ;  in 
which  case,  if  X  represent  the  area  of  a  section  perpendicular  to  this 
axis,  and  x,  its  distance  from  the  plane  yz,  then  will  Xdx,  be  an 
elementary  volume  symmetrically  disposed  in  regard  to  the  axis  x, 
and  Equations  (92),  become 


/x' 
nXx 


d  X 


(107) 


y.  =  0, 

2,  =  0, 
and,  I 

V=f^„Xdx (108) 

Examiile  1. — Find  the  position  of  the  centre  of  gravity  of  a  semi- 
ellipsoid,  the  equation  of  whose  surface  is 

The  axes  of  the  elliptical  section  parallel  to  the  plane  y  z,  are. 


MECHANICS     OF    SOLIDS.  Ill 

whence, 

and,  Equations  (107),  ' 

f"  irBC  (l  -  ^,)  .vdx 

If  the  solid  be  one  of  revolution  about  the  axis  of  .r,  then,  denotinjr 
by 

F{xy)  =  0, (109) 

the  equation  of  the  meridian  section  by  the  plane  a;  y,  -will 
and  Equations  (107)  and  (108),  may  be  written, 


/  ,,    If  y"  X  d  X 


^'  = ' (110) 


V 


v  = 


f],  '^y'^dx (Ill) 


Example  1. — Required   the  position    of  the   centre   of   gravity    of  a 
^Jaraholoid  of  revolution. 

In  this  case.  Equation  (109), 

F{xy)  =  7f  -22)x  =  0, 
whence, 

F  =  2*^3  /   xdx, 

2ifp   Hx'^dx        o 

X.   —  =r  —  a, 

2'rr  p   I    xdx 

J  a 


112  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Example  2. — Required   the  j^osition  of  the   centre   of  gravity   of  the 
volume  of  a  sjjherical  segment  or  zone. 

F{xy)  =  2/2  _|_  a.2  _  ^2  _  0^ 
whence,  for  a  zone, 

F=  -g  r'{cfi  -  x^)dx 

Jx" 

nx' 

*  /  ,.  ("^  ~  x"^)  .X  .dx 


V  ,/("^  —  a;2)  dx 


or, 


and  for  a  segment,  a;"  =z  a, 

3    r,  ,    2a2  -  a;'2"| 

If  the  volume  have  a  plane  face,  and  be  of  such  figure  that  the 
areas  of  all  sections  parallel  to  this  flice,  are  connected  by  any  law 
of  their  distances  fi'om  it,  the  position  of  the  centre  of  gravity,  may 
also  be  found  by  the  method  of  single  integrals. 

Example  1. — Find  the  centre  of  gravity  of  any  j^y^'amid. 

Find  by  the  method  explained,  the  centre  of  gravity  of  the  base 
of  the  pyramid,  and  join  this  point  with  the  vertex  by  a  straight  line. 
All  sections  parallel  to  the  base  are  similar  to  it,  and  will  be  pierced 
by  this  line  in  homologous  points  and  therefore  in  their  centres  of 
gravity.  Each  section  being  supposed  indefinitely  thin,  and  its  weight 
acting  at  its  centre  of  gravity,  the  centre  of  gravity  of  the  entire 
pyramid  will,  §  97,  be  found  somewhere  on  the  same  line. 

Take  the  origin  at  the  vertex,  draw  the  axis  x  perpendicular  to 
the   plane    of    the    base,    and    the    plane   xy    through   its    centre    of 


MEClIxiNICS    OF    SOLIDS. 


113, 


gravity ;  and   let  X  represent  any  section  parallel  to    the   base,  then 
will  Equations  (92)  become, 


xdx 


y,  = 


V 
z^  =0, 


dx 


and, 


„  Xdx. 

Represent  by  A  the   base  of  the  pyramid,  c  its  altitude,  and  let 

7j  =  ax, 

be   the  equation  of  the  line  joining  the  vertex  and  centre  of  gravity 
of  the  base. 
Then, 

^  :  X :  :  c2  :  a;2, 

X  =  — , 


and  for   any  frustum, 


^r  3  7 


'  Ax^dx 


Vj  = 


A 

a  A 
c 


In  x^  dx 

■ — r-  /  rr   X-^  dx 


-— -  /   x~dx 


-  T  \x"i  -  x'J  ' 


T  "  \x"^  -  x'J  ' 


and  for   the   entire  pyramid,  make  x"  =  c,  and  x'  =  0,  which  give 

y,  =  f  a  c ; 


,  IM         ELEMENTS     OF     ANALYTICAL    MECHANICS. 

wlience  we  conclude  that  the  centre  of  gravity  of  a  -pyramid  is  on 
the  line  drawn  from  the  vertex  to  the  centre  of  gravity  of  the  base, 
and  at  a  distance  from  the  vertex  equal  to  threefourths  of  the  length 
of  this   line. 

The  same  rule  obviously  applies  to  a  cone,  since  the  result  is 
independent  of  the  figure  of  the   base. 

The  weight  of  a  body  always  acting  at  its  centre  of  gravity,  and 
in  a  vertical  direction,  it  follows,  that  if  the  body  be  freely  sus- 
pended in  succession  from  any  two  of  its  pomts  by  a  perfectly 
flexible  thread,  and  the  directions  of  this  thread,  when  the  body  is 
in  equilibrio,  be  produced,  they  will  intersect  at  the  centre  of  gravity ; 
and  hence  it  will  only  be  necessary,  in  any  particular  case,  to  deter- 
mine this  point  of  intersection,  to  find,  experimentally,  the  centre 
of  gravity  of  a  body. 

THE   CENTEOBAKYC   METHOD. 

§124. — Resuming   the   second  of  Equations  (95)  and  (103),  which 
are. 


2// 


L'y^'\/^''% 


in  which 


and 


=/;^^V 


1 4-  "^y"" 

ax- 


y,= 


^,fUy"''-y")d=c 


,'hich 


s  =  y //  {y"  —  y')dx; 
clearing    the    fractions    and  multiplying    both   members    by    2*,   we 
shall   have, 

2,i.y,s  =  jl'l'ny    y/dx'  +  JF,      •     •     •     (112) 
^■rty^s^  £',  ':r{y"^  -y'^)dx     ....      (113) 


MECHANICS     OF     SOLIDS. 


115 


The  second  member  of  Equation  (112)  is  the  area  of  a  surface 
generated  by  the  revolution  of  a  plane  curve,  whose  extremities 
are  given  by  the  ordinates  answering  to  the  abscisses  x'  and  x'\ 
about  the  axis  x.  In  the  first  member,  s  is  the  entire  length  of 
this  arc,  and  2*y^  is  the  circumference  generated  by  its  centre  of 
gravity.  Hence,  we  have  this  simple  rule  for  finding  the  area  of  a 
surface  of  revolution,  viz.  : 

Multiphj  the  length  of  the  generating  curve  by  the  circumference 
described  by  its  centre  of  gravity  about  the  axis  of  rotation;  the 
product  will  be    the   required   surface. 

The  second  member  of  Equation  (113)  is  the  volume  generated 
b.y  a  plane  area,  bounded  by  two  branches  of  the  same  curve  or 
by  two  different  curves,  and  the  ordinates  answering  to  the  abscisses 
x'  and  x'\  about  the  axis  x.  s,  in  the  first  member,  is  the  generating 
area,  and  2'ry^  the  circumference  described  by  its  centre  of  gravity. 
Hence,  this  rule  for  finding  the  volume  of  any  surface  of  revolution,  viz. : 

Mulfqyly  the  generating  area  by  the  circumference  described  by  its 
centre  of  gravity  about  the  axis  of  rotation  ;  the  2^1'odnct  xvill  he  the 
volume   sought. 

Example  \.— Required  the  measure  of  the  surface  of  a  right  cone. 

Let  the  cone  be  generated  by  the 
rotation  of  the  line  A  B  about  the 
line  A  C.  The  centre  of  gravity  of 
the  generatrix  is  at  its  middle  point 
(?,  and  therefore,  the  radius  of  the 
circle  described  by  it  will  be  one- 
half  of  the  radius  C  B,  of  the  circu- 
lar base  of  the  cone.      Hence, 

BC.AB 


'Ztiy^.s 


=  If  BC.AB. 


Example  2. — Find  the  volume  of  the  cone. 

The  area  of  the  generatrix  ABC,  is  ^  B  C .  A  C ;   and  the  radius 
of  the  circle  described  by  its  centre  of  gravity  is  ^  B  C.     Hence, 


^nry^s  =  ?j'X  B  C 


BC.A  C 


BC^.AC 


116 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


CENTRE     OF    EST:RTIA. 

S  125. — When  the  elementary  masses  of  a  body  exert  their  forces 
of  inertia  simultaneously  and  in  parallel  directions,  they  must  expe- 
rience equal  accelerations  or  retardations  in  the  same  time,  and  the 
factor 

in  the  measures  of  these  forces,  as  giyen  in  Equation  (18),  must  be 
the  same  for  all.  Substituting  these  measures  for  F',  P",  &c.,  in 
Equations  (G2),  wc  find, 


yy 


dh 

" 

•  2 

m  x' 

dfi 

_  2  m  x'  _ 

d"S 

~     2  ??i    ' 

.  2 

m 

dfi 

dh 

•  2 

m  y' 

dfi 

2  m  y'  _ 

d^s 

~    2  ?yi    ' 

.2 

in 

dfi 

d-'s 

•  2 

m  z' 

dt" 

2  m  z' 

IpT 

2  m 

•  2 

m 

dfi 

(114) 


.  Whence,  Equations  (8G),  the  centre  of  inertia  coincides  with  the 
centre  of  gravity  when  the  latter  force  is  constant,  both  being  at  the 
centre  of  mass.  In  strictness,  however,  the  centre  of  gravity  is 
always  below  the  centre  of  inertia;  for  when  the  variation  in  the 
force  of  gravity,  arising  from  change  of  distance,  is  taken  into 
account,  the  lower  of  two  equal  masses  will  be  found  the  heavier. 
And  in  bodies  whose  linear  dimensions  bear  some  appreciable  propor- 
tion to  their  distances  from  the  centre  of  attraction,  the  distance 
between  these  centres  becomes  sensible,  and  gives  rise  to  some  curious 
phenomena. 


MECHANICS     OF    S(>LIDS. 


117 


MOTION    OP    THE    CENTRE    OF    INERTIA. 

§  126. — Substitute  in  Equations  (A),  the  values  of  d^x,  d^^j,  and  d^z, 
given  by  Equations  (34),  and  \vc  have,  because  dt  \s  constant,  and 
d'^x^,  d'^y^  and  d^z^,  will  each  be  a  common  factor  for  all  the  elemen- 
tary masses, 


2  P  cos  a  —  31- 


dt'^ 


dt- 


2  m  .  d^  x'  =  0, 


d'^  7/  1 

2  P  cos  /3  -  M'  -^ —  .  2  m .  tZ2y'  =  0, 

^  dt^  dC"  ''  ' 


2  P  cos  7  —  J/' 


d'^z, 
dt" 


dl'^ 


Im.d^z'  =  0. 


in  -which  If,  denotes  the  entire  mass  of  the  body,  being  equal  to  2  m. 
Denote  by  .t,  y,  z,  the  co-ordinates  of  the  centre  of  inertia  referred 
to  the  movable  origin,  then.  Equations  (114), 

M.  X  =  Imx', 

M.  z  =  'S.  m  z\ 
and  differentiating  twice, 


M.  d^x  =  2  ?/i .  d^x', 
M.  cZ2y  r=  2  ??i .  dhj\ 
M.  dH  —  2  ??i .  dH\ 


(115) 


which  substituted  in  the   preceding  Equations,  give. 


jP.eosa-il/.^-if._=0, 

^  ,,    d'^z,  ,,     d~z 

jP.eosr-J/-.^-J/-^j-=-.0, 


(116) 


118  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and  if  the    movable    origin    be    taken    at    the   centre    of  inertia,    then 
wiU, 

d^x  r=  0,     dhj  =  0,     d'^z  ^0; 

and   a-^ ,  y^ ,  s^ ,    will  become  the  co-ordinates  of  the  centre  of  inertia 
referred  to  the  fixed  origin,  and  we  have, 

2  P  .  cos  a  —  i/-  -^  =  0, 


2P.C0S/3  -M~  =  0, 
cPz 


(117) 


Equations  which  are  wholly  independent  of  the  relative  positions 
of  the  elementary  masses  m',  m"  &c.,  since  their  co-ordinates  x',  y', 
z',  &c.,  do  not  enter.  It  will  also  be  observed  that  the  resistance  of 
inertia  is  the  same  as  that  of  an  equal  mass  concentrated  at  the 
body's  centre  of  inertia. 

Whence  we  conclude,  that  when  a  body  is  subjected  to  the  action 
of  any  system  of  extraneous  forces,  the  motion  of  its  centre  of  inertia 
will  be  the  same  as  though  the  entire  mass  were  concentrated  into 
that  point,  and  the  forces  applied  without  change  of  intensity  and 
parallel  to  their  primitive   directions,  directly  to  it. 

This  is  an  important  foct,  and  shows  that  in  discussing  the  motion 
of  translation  of  bodies,  we  may  confine  our  attention  to  the  motion 
of  their  centres  of  inertia  regarded  as  material  points. 

KOTATION    AEOUND    THE    CENTRE    OF    INERTIA. 

§  127. — Now,  retaining  the  movable  origin  at  the  centre  of  inertia, 
substitute  in  Equations  (B),  the  values  of  d^.v,  dhj,  and  d-z,  as  given 
by  Equations  (34),  and  reduce  by  the  relations, 

M.dH  —  Sw.t^V  =  0, 
M.  d-^y  =  2  m  .  dhf  -  0, 
M.d-h  =  2  m.c^V  =  0; 


MECHANICS    OF    SOLIDS. 


119 


0, 


and  Ave  have, 

2  P.  (cos  /3  .  x'  -  cos  a  .  y')  +  -  "» •  (^  -^^^  '^        lifi'      ) 

2  P.  (cos  a  .  s'  —  cos  7  .  x')  —  :s  m  ■  {^-^^^  '  d¥  '     )  ~    ' 

2P.(cos7.y-cos/3..')  -2m.  [-        .  y'  - -^'   -z'  "^  =  0; 


(118) 


from  which  all  traces  of  the  position  of  the  centre  of  inertia  have 
disappeared,  and  from  which  we  infer  that  when  a  free  body  is  acted 
upon  by  any  system  of  forces,  the  body  will  rotate  about  its  centre 
of  inertia  exactly  thq  same  whether  that  centre  be  at  rest  or  in 
motion. 

§  12S. — And  we  are  to  conclude,  Equations  (117)  and  (118),  that 
when  a  body  is  subjected  to  the  action  of  one  or  more  forces,  it  will 
in  general,  take  up  two  motions — one  of  translation,  and  one  of  rota- 
tion, each  being  perfectly  independent  of  the  other. 

§129. — Multiply  the  first  of  Equations  (117),  by  y^ ,  the  second  by 
Xf ,  and  subtract  the  first  product  from  the  second ;  also,  the  first  by 
z^  5  the  third  by  x^ ,  and  subtract  the  second  of  these  products  from 
the  first ;  also  the  third  by  y^  ,  and  the  second  by  z^ ,  and  subtract 
the  second  of  these  products  from  the  first,  and  we  have, 


2(Pcos/3).:r,-2(Pcosa).y,-i/.  (^^'X,  -^'l/)  =0, 
2(Pcosa)..-2(Pcos7).:r-J/.  (^..^-^..r)  =0, 
2(Pcos  7).y -2(Pcos/3).. -i/.  (^  •  y,  _  ^ .  .,)  =  0  ; 


(110) 


Equations  from  which  may  be  found   the   circumstances  of  motion 
of  the  centre  of  inertia  about  the  fixed  origin. 


120  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


MOTION    OF    TEANSLATION, 


§  130. — Regarding  the  forces  as  applied  directly  to  the  centre  of 
inertia,  replace  in  Equations  (117),  the  values  2  P.  cos  a,  2  P.  cos /3, 
and  2  P .  cos  y,   by  X,   Y,  and  Z,  respectively,  and  we  may  write, 


Y-M  ^-0 


(120) 


frora  ■which  the  accents  are  omitted,  and  in   which  a*,  y,  and  z,   must 
be  understood  as  appertaining  to  the  centre  of  inertia. 


GENERAL    THEOKEM    OF    WORK,    A^LOCITY    AND    LmNG    FORCE. 

§131. — Multiply  the  first  of  Equations  (120)  by  2fZ.r,  the  second 
by  2o??/,  the  third   by  2  c?  2,  add   and   integrate,  we  have 

/.  dxl    _1_    dill    _|_    (7r2 

2f{Xdx  +  Ydy  +  Zdz)  -  M.  ?.,  +  C  =  0, 


But, 


whence, 


(/^2  '    ~  ~dfi 


dt^ 


=   F2 


^[{Xdx  +  FJy  +  Zdz)  -  i/.  F2  +  (7  =  0 


(121) 


The  first  term  is,  §  101,  twice  the  quantity  of  Avork  of  the  ex- 
traneous forces,  the  second  is  twice  the  quantity  of  work  of  the 
inertia,  measured  by  the  living  force,  and  the  third  is  the  constant 
of  integration. 

If  the  forces  X,  P",  Z,  be  variable,  they  must  be  expressed  in 
functions    of    .r,    y,    2,    before     the     integration     can    be    performed. 


MECHANICS     OF    SOLIDS.  121 

Supposing  this  latter  condition  fullilled,  and  that  the  forms  of  the 
functions  are  such  as  make  tlie  integration  possible,  we  may  write, 

F{xi/z)  -  i-M.V^  -{-  C  =0,      ....     (122) 

and  between  the  limits  x^  y,  z^    and   x/  y/  zf , 

F  {x/  vl  z!)  -  F  {x,  y,  z)  =  i  if  ( F  '2  _  F2)  .     .  (123) 

whence  we  conclude,  that  the  quantity  of  work  expended  by  the 
extraneous  forces  impressed  upon  a  body  during  its  passage  from  one 
position  to  another,  is  equal  to  half  the  difference  of  the  living  forces 
of  the  body  at  these  two  positions. 

We  also  see,  from  Equation  (123),  that  whenever  the  body 
returns  to  any  position  it  may  have  occupied  before,  its  velocity  will 
be  the  same  as  it  was  previously  at  that  place.  Also,  that  the 
velocity,  at  any  point,  is  wholly  independent  of  the  path  described. 

§  132.— If 

Xdx  +   Y d y  -\-  Z d z  =  0, 

the  extraneous  forces  will,  §  101,  be  in  cquilibrio,  and 


M 


=  \/^ 


that    is,   the    velocity   will   be    constant,   and    the    motion,    therefore, 
uniform. 


CENTRAL    FORCES. 

§133. — Forces  which  act  towards  a  given  point,  either  at  rest  or 
in  motion,  and  the  intensities  of  which  depend  upon  the  distance 
from  that  point,  are  called  central  forces.  The  forces  of  nature  are 
of  this   description. 

It  will  always  be  possiljlc  to  find  the  velocity, — that  is,  to  integrate 
the  first  term  in  Equation  (121),  when  the  extraneous  forces  are 
directed  to  fixed  centres,  and  their  intensities  arc  expressed  in  functions 
of  the  body's  distances  from  these  centres. 


123  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


For,  denote  the  constant 
co-ordinates  of  the  fixed  cen- 
tres by  a  b  c,  a'  b'  c,  &c., 
and  the  distances  from  the 
body  to  these  centres  by 
p,  2^'i  *^c.,  then  will 


jr 


■y X 


/      \/ 


X  —  a  y  —  h  z  —  c 

cos  a  =  ,     cos  fd  =  ,     cos  y  =z  ; 


2)  p 

and  the  same  for  the  other  centres,  whence, 


p  p' 


Z  =  P  .  -? ^  +  P'.  iJTA  +  &c. 

P  p' 


Multiplying    the    first   by    dx,    the    second    by  dy,  the    third    by    dz, 
adding  and  integrating,  there  will  result, 


f{Xdx  -f  Ydy  +  Zdz)  =  . 


but, 


whence, 


^  +  &c; 


P  =  V  (-^  -  «)'  +  {y  -  bf  +  {z-  cf, 


dp  =  -—  dx  + .  dy  -\ dz 


P 


P 


P 


MECHANICS    OF    SOLIDS.  123 

and  the  same  fur  d  i)' ;  ^vllich  substituted  in  the  preceding  equation 
for  the  -work,  gives, 

f  {Xdx  +   Ydy  +  Zdz)  =z  f  {Pd2>  +  F'djj'  +  &c.)  ; 

but,  by  hypothesis  F  is  a  function  of  ^j ;  also,  P'  of  j^',  &c. ;  cacli 
tevm  is,  therefore,  a  function  of  a  single  variable;  whence,  the  truth 
of  the  proposition. 

Substituting  in  Equation  (1'31),  we  get, 

f{Pdp  +  P'dp'  +  &c.)  -  iJIV^  +(7=0.     .  (123)' 


STABLE    AND    TNSTABLE    EQUILIBRIUM. 

§  134. — Resuming  Equation  (123),  omitting  the  subscript  accents, 
and  bearing  in  mind  that  the  co-ordinates  refer  to  the  centre  of 
inertia,  into  which  we  may  suppose  for  simplification  the  body  to  be 
concentrated,  we  may  write, 

iMV'^  -  \MV^  =  F{x'y'z')  -  F{xyzl 

in   which 

F{xyz)  =  /{Xdx  +   Ydy  +  Zdz), 
and 

dF{xyz)  =  Xdx  +  Ydy  +  Zdz. 

Now,  if  the  limits  x'  y'  z'  and  x  ?/  2  be  taken  very  near  to  each 
other,  then    will 

x'  =  X  4-  dx;     y'  —  y  -\-  dy\     z'  =  s  +  dz; 

which   substituted    above,  give 

|3/F'2  _  iMV^  =  F{x  +  dx,  y  -{-  di/,  z  +  dz)  -  F{xyz), 

and  developing   by  Taylor's  theorem, 

Adx  +  J]dy  +  Cdz 


^  ^  '  +  A' d  x^  -\-  B' d  y^~  +  C  d  z"^  -{-  B, 

in    which    Z>   denotes    the    sum    of    the    terms    involving    the   higher 
powers    of  d x,  dy  and  dz^  (.->■>./  dx  /y     cl/dt     c^i    It. 


124  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

If  \  M  V^  be   a   maximum  or    minimum,  then    will 

Adz  +  Bdy  +  Cdz  =  0; (123)" 

and   since 

Adx  -{-  Bdy  +   Cdz  =  dF{xyz)  =  Xdx  +  Ydy  +  Zdz, 

we   have, 

Xdx  +  Ydy  +  Zdz  =z  0. 

But  when  this  condition  is  fulfilled,  the  forces  will,  Equation  (CO), 
be  in  equilibrio ;  and  we  therefore  conclude  that  whenever  a  body 
whose  centre  of  inertia  is  acted  upon  by  forces  not  in  equilibrio, 
reaches  a  position  in  which  the  living  force  or  the  quantity  of 
work  is  a  maximum  or  minimum,  these  forces  will   be   in  equilibrio. 

And,  reciprocally,  it  may  be  said,  in  general,  that  when  the  forces 
are  in  equilibrio,  the  body  has  a  position  such  that  the  quantity  of 
action  will  be  a  maximum  or  minimum,  though  this  is  not  always 
true,  since  the  function  is  not  necessarily  either  a  maximum  or  a 
minimum  when  its  first  differential  co-efficient  is  zero. 

§  135. — Equation  (123)",  being  satisfied,  we  have 

|3/F'2  _  ^MV^  =  ±  {A'dx^-  +  B'dy^  +  C'dz-^  +  B)  .  (124) 

The  upper  sign  answers  to  the  case  of  a  minimum,  and  the  lower 
to  a  maximum. 

Now,  if  V  be  very  small,  and  at  the  same  time  a  maximum,  V 
must  also  be  very  small  and  less  than  V,  in  order  that  the  second 
member  may  be  negative ;  whence  it  appears  that  whenever  the  system 
arrives  at  a  position  in  which  the  living  force  or  quantity  of  work  is 
a  maximum  and  the  system  in  a  state  bordering  on  rest,  it  cannot 
depart  far  from  this  position  if  subjected  alone  to  the  forces  which 
brought  it  there.  This  position,  which  we  have  seen  is  one  of  equi- 
librium, is  called  a  position  of  stable  equilibrium.  In  fact,  the  quantity 
of  work  immediately  succeeding  the  position  in  question  becoming 
negative,  shows  that  the  projection  of  the  virtual  velocity  is  negative, 
and  therefore  that  it  is  described  in  opposition  to  the  resultant  of  the 
forces,  which,  as  soon  as  it  overcomes  the  living  force  already  existing, 
will  cause  the  body  to  retrace  its  course. 


MECHANICS     OF    SOLIDS.  125 

gl3(j, — If^  on  the  contrary,  the  body  reach  a  position  in  which  the 
quantity  of  work  is  a  minimum,  the  upper  sign  in  Equation  (I'i-l), 
must  be  taken,  the  second  member  will  always  be  positive  and  there 
will  be  no  limit  to  the  increase  of  V.  The  body  may  therefore 
depart  further  and  further  from  this  position,  however  small  V  may  be ; 
and  hence,  this  is  called  a  position  of  unstable  equilibrium. 

R  137. — If  the  entire  second  member  of  Equation  (124),  be  zero, 
then  will, 

^3/7'2  _  ^jj/F2  =  0, 

and  there  will  be  neither  increase  nor  diminution  of  quantity  of  work, 
and  whatever  position  the  body  occupies  the  forces  will  be  in  equili- 
brio.     This  is   called   equilibrium  of  indifference. 

g  138. — If  the  system  consist  of  the  union  of  several  bodies  acted 
upon  only  by  the  force  of  gravity,  the  forces  become  the  Aveigbts 
of  the  bodies  which,  being  proportional  to  their  masses,  will  be  con- 
stant. Denoting  these  weights  by  W,  W'\  W",  &c.,  and  assum- 
ing the  axis  of  z  vertical,  we  have  from  Equations  (86), 

Rz^  =  W'z'  +  W"  z"  +  W"'z"'  +  &c., 

in  which  i2,  is  the  weight  of  the  entire  system,  and  z^  the  co-ordi- 
nate of  its  centre  of  gravity;    and  differentiating, 

Rdz,  =  Wdz'  +  W'dz"  -\-  W"'dz"'  +  &c.    .     .     .    (125) 

Now,  if  z^  be  a  maximum  or  minimum,  then  will 

W'  dz'  +   W'dz"  +   W'dz'"  +  &c.  =  0, 

which  is  the  condition  of  equilibrium  of  the  weights.  Whence,  we 
conclude  that  when  the  centre  of  gravity  of  the  system  is  at  the 
highest  or  lowest  point,  the  system  will  be  in  e(iuilibrio. 

In  order  that  the  virtual  moment  of  a  weight  may  be  positive, 
vertical  distances,  when  estimated  downwards,  inust  be  regarded  as 
positive.  This  will  make  the  second  differential  of  z^ ,  positive  at 
the  limit  of  the  highest,  and  negative  at  the  limit  of  the  lowest 
point.  The  equilibriuiu  will,  therefore,  be  stable  when  the  centre  of 
gravity  is  at  the  lowest,  and  unstable  when  at  the  highest  point. 


126  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Integrating  Equation  (125),  between  the  limits  z^  =  H^  and 
z^  =  H',  z'  —  111  and  z'  —  h\  &c.,  and  we  find, 

R{H,  -  H')  =  W'{h,  -  h')  +  W"  {h,,  -  h")  +  &c. ;  .  (126) 

from  which  we  see  that  the  work  of  the  entire  weight  of  the  system, 
acting  at  its  centre  of  gravity,  is  equal  to  the  sum  of  the  quantities 
of  work  of  the  component  weights,  which  descend  diminished  by  the 
sum  of  the  quantities  of  work  of  those  which  ascend. 

rNITIAL    COKDITIONS,    DIRECT    AND    KEVEESE    PKOBLEM. 

§  130. — By  integrating  each  of  Equations  (120)  twice,  we  obtain 
three  equations  involving  four  variables,  viz.  :  x,  y,  z  and  t.  By 
eliminating  t,  tliere  will  result  two  equations  between  the  variables 
.r,'  y  and  z,  which  will  be  the  equations  of  the  path  described  by 
the   centre    of  inertia  of  the   body. 

§  140. — In  the  course  of  integration,  six  arbitrary  constants  will 
be  introduced,  whose  values  are  determined  by  the  initial  circum- 
stances of  the  motion.  By  the  term  initial,  is  meant  the  epoch 
from   which   t  is    estimated. 

The  initial  elements  are,  1st.  The  three  co-ordinates  which  give 
the  position  of  the  centre  of  inertia  at  the  epoch ;  and  2d.  The 
component  velocities  in  the  direction  of  the  three  axes  at  the  same 
instant. 

The  general  integrals  determine  the  nature  only,  and  not  the 
dimensions    of  the   path. 

§  141. — Now  two  distinct  propositions  may  arise.  Either  it  may 
be  required  to  find  the  path  from  given  initial  conditions,  or  to 
find   the    initial   conditions    necessary    to   describe   a  given   path. 

In  the  first  case,  by  difierentiating  the  three  integrals  with  respect 

dx     dxi     dz 

to    t,    we   obtain    three    equations    involving    .r,  y,  0,    y-)    — '    -i-'    i, 

and  the  arbitrary  constants ;  making  t  equal  to  zero,  and  giving 
the   initial    elements    their    values,   there  will   result  three  more  equa- 


MECHANICS     OF     SOLIDS. 


12T 


tions  involving  the  arbitrary  constants  and  known  quantities.     From 

these    six    equations    we    may    find    the   arbitrary    constants,    and    the 

problem   is   completely  solved. 

In   the   second  case,  we  shall  have  given  two  equations  involving 

fZ  X     up"  ?/       c?^  z 
X.  y,  z.  from  which  may  be  found  -—ri   -Hr'   —r-r,->   or  X.  Y.  Z,  which 
'  •^'    '  ^  ill-      dfi      dt^  J      )     5 

shows   that   the   problem   is  indeterminate. 

But   Equation   (121)  being    differentiated   and  divided   by  the   dif- 
ferential  of  one   of  the   variables,  say  dx,  gives 


d  X  dx  dx 


(127) 


which  is  a  third  equation  involving  X,  Y,  Z,  and  V.  By  assuming 
a  value  for  any  one  of  these  four  quantities,  or  any  condition  con- 
necting them,  the  other  three  may  be  found  in  terms  of  x,  y  and  z. 


VERTICAL   MOTION   OF   ITEAYY   BODIES. 

§  142. — When  a  body  is  abandoned  to  itself,  it  falls  toward  the 
earth's  surface.  To  find  the  circumstances  of  motion,  resume  Equa- 
tions (120),  in  which  the  only  force  acting,  neglecting  the  resistance 
of  the  air,  will  be  the  weight  =  Mg ;  and  we  shall  have,  Equa- 
tions (in), 

2  P  cos  a  =  X  =  Mg  .  cos  a  ; 
2  P  cos  /3  =  Y  =  Mg  .cos(5; 
2  P  cos  y  =.  Z  zzz  Mg  .  cos  y ; 

in  which  M  denotes  the  mass  of 
the  body.  The  force  of  gravity 
varies  inversely  as  the  square  of 
the  distance  from  the  centre  of 
the  earth,  but  within  moderate 
limits  may  be  considered  invaria- 
ble. The  weight  will  therefore  be 
constant  during  the  fall. 

Take  the  co-ordinate  z  vertical, 
and  positive  when  estimated  downwards,  then  will 


cos  a  =  0  ;     cos  ,S  r=  0  ;     cos  v  =  1  ^ 


128  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and  Equations  (120)  become,  after  omitting   the  common   factor  M^ 


and  integrating, 


dx  dy 

—-  =  u   •   -—  =  u   : 

d  t         ^^   dt         y 


dz  ,       , 

—  ^  V  =fjt^n^_ (128) 

in  which  v  is   the  actual  velocity  in   a  vertical   direction. 
Making  ^  =  0,  -vve   have 

dz  _ 

'd't~     ' 

\ 
The   constants    u  ,  u    and  u  .    are    the    initial    velocities    in    the 

directions   of  the   axes  .r,  y  and  2,  respectively.     Supposing   the  first 
two   zero,  and   omitting   the   subscript   2',  from  the   third,  we   have, 

dx        „      dii 
dt-'-^tt-'-^ 

v=-^  =  ^/  +  M (129) 

Integrating  again,  we  find 

x^  C;     y  =  C", 
z^lgt-^  A^  ut-^  C", 

and  if,  when    t  —  0,  the   body  be   on   the  axis  z,  and   at  a   distance 
below   the   origin    equal    to    a,    then    will 

a;  =  0;     y  =  0  ; 
z  z=  lyt"  +  ut -\-  a (130) 

If  the  body  had  been  moving  upwards  at  the   epoch,  then  would 
u   have   been    negative,  and.  Equations  (129)  and  (130), 

V  z=  gi  —  u (131) 

z=lfff-  —  nt-j-a (132) 


MECHAXICS     OF    SOLIDS.  129 

If  the  body  had  moved  from  rest  at  the  epoch  and  from  the 
origin  of  co-ordinates,  then  would  v  be  the  actual  velocity  generated 
by  the  body's  weight,  and  z  =  h,  the  actual  space  described  in  the 
time  t;  and  Equations  (129)  and  (130)  would  become, 

V  —  gt (133) 

h  =  yp (134) 

and  eliminating  /, 

V  =  y/~Tfk (135) 

whence,  we  see  that  the  velocity  varies  as  the  time  in  which  it  is 
generated ;  that  the  height  fallen  through  varies  as  the  square  of  the 
time  of  foil ;  and  that  the  velocity  varies  directly  as  the  square  root 
of  the  height. 

The  value  of  A,  is  called  the  height  due  to  the  velocity  v  ;  and 
the  value    v,  is  called  the  velocity  due  to  the  height  )i. 

If,  in  Equation  (132),  we  suppose  a  =  0,  we  shall  have  the  case 
of  a  body  thrown  vertically  upwards  with  a  velocity  m,  from  the 
origin,  and  we  may  write, 

V  z=z  gt  —  n (136) 

z  —  ^gf^  —  ut (137) 

when  the  body  has  reached  its  highest  point,  v  will  be  zero,  and  we 
find, 

g  t  —  ?^  =  0  ; 
or, 

t  —  — ; 

9 

which  is  the  time  of  ascent;  and  this  value  of  C,  in  Equation  (134), 
will  give  the  greatest  height,  h  =  z,  to  which  the  body  will  attain, 

^'    =    -^g (138) 

§  143. — In  the  preceding  discussion,  no  account  is  taken  of  the 
atmospheric  resistance.      For  the  same  body,  this  resistance  varies  as 

9 


130  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

the  square  of  the  velocity,  so  that  if  h,  denote  the  velocity  Avhen  the 
resistance  becomes  equal  to  the  body's  weight,  then  will 

M .  g  .v^ 

be  the  resistance  when   the   velocity  is  v,  and  in  Equations  (117),  we 
shall  have, 

2  P  cos  cc  =  A''  =  M  y  cos  a  +  Mg  •  —  •  cos  a', 

2Pcos/3  =  Y  =  M g  cos/3  +  Mg  •  t^-  cos  /3', 

"  2  P  cos  7  =  Z  =  M g  cos  y  +  M g  •  —  •  cos  7' ; 

taking  the  co-ordinate  z,  vertical  and  positive  downward,  then  will, 

cos  a  =  cos  a'  =  0, 
cos  /3  =  cos  /3'  =  0, 
cos  7  =  1,    cos  7'  =  —  1  ; 

and,  supposing  the  body  to  move  from  rest.  Equations  (120),  give, 
d"  z  V" 

1     .       d'^  z  .      ,  dv 

Omitting  the  common  fictor  i/,  and  replacuig  -— ^  by  its  value  —  , 


dv 

9 


('-0 


dt     ■"  \       k^ 

whence, 


k^.dv     _Jc_r     dv     ^__dv_\     .     .     (139) 
"  ^  -   yl.2  _  ^2    -    2    V  yt  +  V  ^     A;  -  V  /  ^       ^ 


Integrating  and  supposing  the  initial  velocity  zero. 


gt^^kAog."^ (140) 


MECHANICS     OF     SOLIDS.  131 

which  gives  the  time  in  terms  of  the  velocity,  or  reciprocally, 

^■  +  ''  =/'^ (141) 


k  —  V 


ill   which  e,  is   the  base  of  the    Naperian   system  of  logarithms,  and 
from  which  we  find, 

.  =  ikiZLf_JiZ_, (142) 

e  A  +  e    I' 
which    gives,  the  velocity    in    terms  of  the   time.     Substituting  for  v, 
its  value   — >     integrating   and   supposing   the   initial   space   zero,    we 
have  I'Wd 

.  =  '^.logl(/^~  +  e"t) (143) 

Multiplying   Equation   (139)    by 

dz 

we  have, 

Jc^.  V  .dv 


gdz  = 


/1-2  -  v2 
and   integrating,  observing   the   initial    conditions   as    above, 

2y       °  k^  —  v^ 

which   gives    the   relation   between  the    space   and    velocity. 

-.11  '' 

As    the    time   increases^  the    c^uantity  e     ''    becomes  less  and  less, 

and    the   velocity.    Equation    (142),    becomes    more    nearly   uniform  ; 

for,  if  t  be   infinite,  then  will 

_  ^ 
e    ~^  =  0, 

and,   Equation   (142), 

V  =  k ; 

making   the   resistance   of  the   air   equal   to   the   body's   weight. 


132  ELEMENTS     OF    ANALYTICAL    MECHANICS.   * 

§  144. — If    the   body    had    been  moving    upwards   with   a    velocity 
V,   then  would  Equations  (120), 


^;±^-M,-M^ 


d  V  u     Z 

substituting   —    for   -r-v  and  omitting  the  common  factor,  we  find, 

do  CI/    t 


k .dv  g  d  t 


(145) 


¥■  +  ^;2  k    ' 

integrating, 

and   supposing   the    initial  velocity  equal  to  a,  we  find 

C  =  tan  .  —5 

rC 

and, 

tan     _  =  tan     -  -  ■- (146) 

Taking  the   tangent   of  both  members  and  reducing,  we  find 

a  —  k  .  tan  — 

v  =  k (147) 

k  -\-  a.  tan  -r- 
k 

which  may  be  put  under  the  form, 

»                                 gt-.gt 
a .  cos  — k .  sm  — 

v  =  k -^ "-      ....     (148) 

.    g t       ,         gt 

a  .  sm  ^ — [-  k .  cos  -;- 

k  k 

Substituting    for  v    its  value    -^i     integrating,    and    supposing    the 
initial  space  zero,  we  have 

,=  log(       .sm^  +  cos-j.     .     .     .    (149) 


"      MECHANICS    OF    SOLIDS.  133 

Multiplying  Equation  (145),  by 


and  we  have, 


dz 


I"  .V  .dv 

9'dz  =  -  J,.,  _^  ,.,-; 


/*=-—.  log 


2^     ^  F  -  «'2 

and  placing  this  value  of  h  equal   to   that  given   by  Equation  (151), 
there  will  result, 


132  ELEMENTS     OF    ANALYTICAL    MECHANICS.   * 


§  144. — If    the   body   had   been   moving   upwards   with   a    velocity 
v,   then  would  Equations  (120), 


■KT    ^^^  1/  i,r  9'"'^ 


dv 


substituting  —    for  -jw'  ^^^^  omitting  the  common  factor,  we  find, 


A. 


t)t^t,^  e^ 


^ 


-^5w« 


^  - 


'^T^^ 


^^  A.  ±n^:^SL 


X^^  ^fc,.  ^ 


/^^/y 


/ 


^    ^/^/VW/^„ 


^  - 


^ 


^ 


^-5'  ^--r.^f^^r  =  ^  ^/^./=^^^.A^ 


a  .  sin  ^  +  A; ,  cos  'V- 
A;  A- 

Substituting   for  v   its  value   — ^?    integrating,    and    supposing    the 


initial  space  zero,  we  have 
^•2 


k"     .       /  a        .     at  qt\ 

^  _.log  (^_.sm-  +  cos-; 


(149) 


V 

MECHANICS    OF    SOLIDS.  133 


Multiplying  Equation  (145),  by 

dz 

and  -sve  have, 

k"^  .V  .dv 


g  .dz  =  — 


F  +  ?;2    ' 


and  integrating,  with  the  same  initial  conditions  of  v  being  equal  to 
a,  when  z  is  zero,  there  will  result, 

;j  ^  ii  .  log  j^l±i^; (150) 

§  145. — If  we  denote  by  /t,  the   greatest  height  to  which  the  body 
will  ascend,  we  have  z  =  h,  when  v  =::  0,  and  hence. 

Finding  the  value  of  /,  from  Equation  (14G),  we  have, 

t  =  —  {  tan tan      -^  )    •     •     •     •     (lo2) 

g    ^  k  k  y 

from  which,  by  making  t;  =  0,  we  have, 

t    =  —  ■  tan~^  4-        (1^3) 

g  k  ^       ^ 

which  is  the  time  required  for  the  body  to  attain  the  greatest  eleva- 
tion. Having  attained  the  greatest  height,  the  body  will  descend,  and 
the  circumstances  of  the  fall  will  be  given  by  the  Equations  of  §  143. 
Denoting  by  a',  the  velocity  when  the  body  returns  to  the  point  of 
starting.  Equation  (144),  gives, 

k-^  F 

II  =  — —  •  l02 


2<7         '°   k"  —  a'2 

and  placing  this  value  of  li  equal   to   that   given   by  Equation  (151), 
there  will  result, 

k"^  _    r-  +  a^ 

k^  -  a'2    ~  ^2       ' 


134  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

whence, 

a'2  =  a?  • 


a2  +  A-2   ' 

that    is,  the  velocitj  of    the  body  when    it    returns    to    the    point    of 
departure  is  less  than  that  with  which  it  set  out. 
Making  v  =  a'  in  Equation  (140),  we  have, 

h      ,       k  +  a' 

and,  substituting  for  a',  its  value  above, 

i,  =  ——  .  log  — 5  •     •     •     •     (154) 

a  value  very  different  from  that  of  ^^,   given  by  Equation  (153),  for 
the  ascent. 

Multiplying  both  numerator  and  denominator  of  the  quantity  whose 
logarithm  is  taken,  by  -y/  o?  -{-  k'^  —  a^,  the  above  becomes, 

tf=-'\og-—=^ (155) 

Adding  Equations  (153)  and  (155),  we  have, 

X-   r      -1   a     ,    ,  h  -1 

t    ^  t,  —  —    tan      -T-  4-  log  : 

or,  making  t  =  f^  +  t  , 

91  ^  tan"'  ^  +  log —-. •     •     •     (156) 

If  a  ball  be  thrown  vertically  upwards,  and  the  time  of  its 
absence  from  the  surface  of  the  earth  be  carefully  noted,  t  will  be 
known,  and  the  value  of  k  may  be  found  from  this  equation.  This 
experiment  being  repeated  with  balls  of  different  diameters,  and  the 
resulting  values  of  k  calculated,  the  resistance  of  the  air,  for  any 
given  velocity,  will  be  known. 


MECHANICS     OF     SOLIDS. 


135 


PKOJECTILES. 

g  14(j. — Any  body  projected  or  impelled  forward,  is  emailed  a  pro- 
jectile^ and  the  curve  described  by  its  centre  of  inertia,  is  called  a 
trajectory.  The  projectiles  of  artillery,  which  are  usually  thrown  with 
great  velocity,  will  be  here  discussed. 

g  147. — And  first,  let  us  consider  what  the  trajectory  w^ould  be 
in  the  absence  of  the  atmosphere.  In  this  case,  the  only  force  which 
acts  upon  the  projectile  after  it  leaves  the  cannon,  is  its  own  weight ; 
and,  Equations  (HT), 

2  P  cos  a  =3  X  =  Mg  cos  a, 
2  P  cos  /3  =  F  =  Mg  cos  /3, 
2  P  cos  7  =z   Z  —  Mg  cos  7. 


Assuming  the  origin 
at  thfe  point  of  de- 
parture, or  the  mouth 
of  the  piece,  and 
taking  the  axis  z 
vertical,  and  posi- 
tive upwards,  then 
will 


cos  a  =  0  ;    cos  ,8  =  0 ;    cos  7  =  —  1  ;    and.  Equations  (120), 
J/.-^=0;    J/.-^  =  0;    M--^=  -Mg; 


dC-  '  d  i 

and  integrating,  omitting  M, 
d  y 


dt'' 


d  X  di/  dz  ^    , 

— —  =  «.    :     — -^  —  u    :     — —  =  —  g  t  +  u 
d  t  "='       dt  y'  .    d  t  "^  ' 


(157) 


Integrating  again,  and  recollecting  that  the  initial  spaces  are  zero,  we 
have, 

x  =  u^-t;    y  =  u^-t;    z=  -^gf^  +  u^'t     •     -(158) 


loG  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and  eliminating  t,  from  tlte  first  two,  we  obtain, 

u 

X 

which  is  the  equation  of  a  right  line,  and  from  which  we  see  that 
the  trajectory  is  a  plane  curve,  and  that  its  plane  is  vertical. 

Assume  the  plane  zx^  in  this  plane,  then  will  y  =  0,  and  Equa- 
tions (158),  become, 

X  =  u^-t;    z=  -^gf'  -^  u^-t.      •     •     •      (159) 

Denote  by  V,  the  velocity  with  which  the  ball  leaves  the  piece, 
that  is,  the  initial  velocity,  and  by  a,  the  angle  which  the  axis  of  the 
piece  makes  with  the  axis  .r,  then  will, 

V  cos  a,    and    V  .  sin  a, 

be  the  lengths  of  the  paths  described  in  a  unit  of  time,  in  the  direc- 
tion of  the  axes  x  and  z,  respectively,  in  virtue  of  the  velocity  V  ; 
they  are,  therefore,  the  initial  velocities  in  the  directions  of  these 
axes;   and  we  have, 

u    =  Fcos  a  ;     u    =  V.  sin  a  • 

which,  in  Equations  (159),  give 

a;=F.  cos  a.  <;     z  —  -  \  g9  Ar  V .■&\\\  a  A      -     •     (1G(  ) 

and   eliminating  t,  we  find 

ff  .x^ 

2  =  a;  tan  a  —   „  „9 2~"  ' 

2  F 2 .  cos^  a 

or    substituting  for    F  its  value   in  Equation  (135), 

z  =  X  tan  a  —  -— 5— (1^1) 

4  h  .  COS''  a 

which  is  the  equation  of  a   parabola. 


MECHANICS     OF    SOLIDS. 


137 


§  148. — The  angle  a  is 
called  the  angle  of  projec- 
tion ;  and  the  horizontal 
distance  A  B,  from  the 
place  of  departure  xi,  to 
the  point  Z),  at  which  the 
projectile  attains  the  same 
level,  is  called  the  range. 

To  find  the  range,  make  s  =  0,  and  Equation  (IGl)  gives 

.r  =z  0,    and    a:  =  4  A  sin  a  cos  a  =  2  A  sin  2  a, 
and   denoting   the  range  by  R, 

i2  =  2  /i .  sin  2  a 


(1G2) 


the   value    of  which  becomes   the   greatest    possible   when   the   angle 
of  projection   is   45°.     Making  a_=  45°,  we   have 


R  =  2h 


(1G3) 


that  is,    the   maximum    range   is   equal   to   twice   the   height  due   to 
the   velocity  of  projection. 

From  the  expression  for  its  value,  we  also  see  that  the  same 
range  will  result  from  two  different  angles  of  projection,  one  of  which 
is   the   complement  of  the   other. 

§  149. — Denoting  by  v  the  velocity  at  the  end  of  any  time  t,  we 
have, 

2  _   ^^*^  _  ^^^  +  dx^ 

or,  replacing  the  values  of  dz  and  dx,  obtained  from  Equations  (160), 

^2  =  F2  -  2  F.^.^sina  +  ^-^    ....     (ig4) 

and   eliminating   t^    by    means    of  the    first   of  Equations    (100),    and 
replacing   V,  in   the  last  term   by  its  value  2^  A, 


^2  _  ^7-2  _  2  ^ .  tan  o- .  X  -\-  g  •  - 


2  h  .  cos2  a 


(165) 


138  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

in  which,  if  we   make   x  =  4  A .  sin  a  cos  a,  we   have   the  velocity   at 
the   point  D, 

v^  =  V\ 

which  shows  that  the  velocity  at  the  furthest  extremity  of  the  range 
is   equal    to    the   initial   velocity. 

Differentiating  Equation  (161),  we  get 

•  ^'  =  tan  ^  =  tan  a  -  ^,     ""    ,       ....     (166) 

dx  2/4.cos2a 

in  which   ^    is   the   angle  which  the   direction  of  the  motion   at   any 
instant  makes   with   the  axis   .r. 
Making   tan  ^  =  0,  we  find 

X  =  2  A  .  cos  a  .  sin  a, 
which,  in  Equation   (161),  gives 

z  =  k.  sin^  a, 

the  elevation  of  the  highest  point. 

Substituting  for  x,  the  range,  4  h  cos  a  sin  a,  in   Equation  (166), 

tan  &  —  —  tan  a, 

which    shows   that   the   angle  of  fall  is  equal  to  minus  the  angle  of 
projection. 

I  150._The  initial  velocity  V  being  given,  let  it  be  required  to 
find  the  angle  of  projection  which  will  cause  the  trajectory  to  pass 
through   a   given  point  whose   co-ordinates  are  a;  =  a  and   z  =  b. 

Substituting   these  in  Equation   (161),  we  have 


b  =  a  tan  a  — 


ff 


4  h .  cos- a 


from   which   to    determine   a. 
Making   tan  cc  =  (p,  we   find 

1 

C0S2  a    _ -, 

1+9 


MECHANICS     OF     SOLIDS. 


139 


which   in  the   equation    above,    gives 

4  /i .  i  +  a2  —  4  A  .  a  .  9  +  a2  ^2  _  0  ; 


whence, 


2/i 


o  =  tana  -  —  ±.4/4  A-  -  4A6  —  a2  .     .     .     (167) 


The  double  sign  shows  that  the  object  is  attained  by  two  angles, 
and  the  radical  shows  that  the  solution  of  the  problem  will  be 
possible  as   long   as 

4  7^2  >  4  /i  I,  4-  (,2. 

Making, 

4  F  —  4  A .  i  —  ft2  —  0,  . 

the  question  may  be  solved  with  only  a  single  angle  of  projection. 
But  the  above  equation  is  that  of  a  parabola  whose  co-ordinates  are 
a  and  b,  and  this  curve  being  con- 
structed and  revolved  about  its  vertical 
axis,  will  enclose  the  entire  spaSe 
within  which  the  given  point  must  be 
situated  in  order  that  it  may  be  struck 
with  the  given  initial  velocity.  This 
parabola  will  pass  through  the  farthest 
extremity  of  the  maximum  range,  and 
at  a  height  above  the  piece  equal  to  h. 


Jl 


J) 


§151. — Thus  we  see  that  the  theory  of  the  motion  of  projectiles 
is  a  very  simple  matter  as  long  as  the  motion  takes  place  in  vacuo. 
But  in  practice  this  is  never  the  case,  and  where  the  velocity  is  con- 
siderable, the  atmospheric  resistance  changes  the  nature  of  the  tra- 
jectory,  and  gives  to  the  subject  no  little  complexity. 

Denote,  as  before,  the  velocity  of  the  projectile  when  the  atmos- 
pheric resistance  equals  its  weight,  by  k,  and  assuming  that  the 
resistance  varies  as  the  square  of  the  velocity,  the  actual  resistance 
at  any  instant  when  the  velocity   is  i',  will  be, 


■^.(/■l'2 

A?2 


=  J/ct'2, 


140  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

by  making, 

Tlie  forces  acting  upon  the  projectile  after  it  leaves  the  piece 
being  its  weight  and  the  atmospheric  resistance,  Equations  (120), 
become, 

M  • =  M  (J  .  cos  a  +  Mc .  V" .  cos  a', 

M-  '-r^  =  Mg  .  cos  /3  +  3fc  .  v^ .  cos  5'," 
a  r 

d?'  z 
M '  -^— r-  =  Mg  .  cos  y  +  Mc .  v^ .  cos  y'. 

Taking  the  co-ordinates    z   vertical,    and    positive    -when   estimated 
u  J)  wards, 

cos  a  —  0;     cos  /3  =  0 ;     cos  y  =  —  1, 

and  because  the  resistance  takes  place  in  the  direction  of  the  trajec- 
tory, and  in  opposition  to  the  motion,  if  the  projectile  be  thrown  in 
the  first  angle,  the  angles  a',  /3',  and  y',   will  be  obtuse, 

,  dx  r,,  dy  ,  dz 

cos  a'  — —  ;     cos  p    = T—  ;     cos  y    = — , 

ds  ds  '  d  s 

and  the  equations  of  motion  become,  after  omitting  the  common 
factor  M, 

d?  X  f/  X 

~d^  ^  ~  ^  ""  '  ~ds  ' 

d^l/  r,     dy 

-^  =  -  C'V--j~; 

Pz  ^     dz 


From  the  first  two  we  have,  by  division, 

d'^y         d"^  X 
dy  d  X  ^ 


MECHANICS    OF    SOLIDS.  lil 

and  by  integration, 

log  d^j  =  log  dx  +  C ; 

and,  passing  to  the  quantities, 

dy  ^  Cd X.  0 

Integrating  again,  we  have, 

y  =  Cx-h  C; 

in  which,  if  the  projectile  be  thrown  from  the  origin,  C  —  0,  thus 
giving  an  equation  of  a  right  line  through  the  origin.  Whence  we 
see  that  the  trajectory  is  a  plane  curve,  and  that  its  plane  is  vertical 
through  the  point  of  departure. 

Assuming    the   plane   z  a*,  to    coincide   with    that  of  the   trajectory, 
and  replacing  v^,  by  its  value  from  the  relation, 

ds^ 


dfi 


we  have, 


d^x 

ds 

d  X 

df^  ~ 

—  c  • 

dt 

dt  ' 

d'^z 

ds      a 

de~ 

-9 

—  c- 

d  I      c 

From  the  first 

we  have. 

d^x 
dfl 

ds 

C  '  ■ 

d  X 

dt 

and  by  integration, 


dt 


dx 


loff  •  -nrr-  =  —  c  .  s  +  C. 
dt 


(1G8) 


Denoting  by  e,  the    base  of  the    Napcrian    system    of    logarithms, 
and  making   0  =  log  ^i,  the  above  may  be  written, 

log  •  -,-    =  —  c .  5  X  log  e  +  log  A, 


142  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

and  passing  from  logarithms  to  the  quantities, 

^=A.e  (169) 

Denoting  by   F,  the  initial   velocity,  and   by  a,  the   angle  of  pro- 
jection, we  have,  by  making  a  =  0, 

—r-  =  A  =   V  cos  a, 
dt  ' 

which   substituted   above,  gives 

dr  — «* 

^  =  F.cosa.e  (170) 

dt  ^ 

To   integrate  the  second   of  Equations  (168),  make 

^=».^, (171) 

dt       ^     dt  ^       ' 

in  "which  p  is   an  additional   unknown    quantity. 

Differentiating    this    equation,     dividing     by    dt,    and    eliminating 

from    the   result,   --^>    by    its    value   in    the   first  of  equations  (168), 

we  have, 

d'^z         dp     dx  ds  dx 

Jfi    ~  dTt'  d~t~  ^^'^  '  dt'  di 

and    substituting    this   value    in    the    second   of   Equations    (168),    we 
have,  after  eliminating    —  by  its  value,  obtained  from  Equation  (171), 

%■'{-,  =  -9 (ira) 

dt     d  t 

and    dividing   this  by   the  square    of  Equation   (170), 


dp 

dt_  g 

dx  ~         V^  cos^  a 
Tt 


(173) 


MECHANICS    OF    SOLIDS.  14:3 

but  regarding  z   and  'p  as  functions  of  x,  we   have,  Ec^uation  (171), 


and, 


dz 

dt_    dz 
•'    ~"  dx         dx 
dTt 


dp 

d  t         dp 

d  X         dx 


(174) 


whence,  makmg   V-  =  2^A,  Equation  (173)  becomes 


(175) 


2cs 


cos^  a 


;  (176) 


in  which  C  is  the  constant  of  integration  ;  to  determine  which,  make 
5  =  0;  this  gives  p  z=  tan  a;  and 

C  = TT-  +  tan  a  .  i/T+lan^a  +  ]oa;  (tan  a  +  ^l-\-t!xn^a)  •  (177) 

2c  h  cos2  a 

From   Equation  (175)  we   have, 

—  2  c  s 

dx  =  —  2h.  cos^  a  .e     •      d])  ; 


from   Equation   (171), 

dz  =:  jJ  •  d X  ] 


142  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and  passing  from  logarithms  to  the  quantities, 

^=A.e  (169) 

Denoting  by   V,  the  initial   velocity,  and   by  a,  the   angle  of  pro- 
jection,  we  have,  by  making  s  =  0, 

— —  =  A  =    V  cos  a 
dt 

which   substituted   above,  gives 

rJr  — <=* 

_  —  jr  r>r,^»  ^  n'y^^ 


To   integrate 


in  which  »  is   a  ^-'/''^  ^     ,  ^    ,  -,  , 

Differentiatin      ^^  ^T^  -    ""^  =  —Jlt—^'^^  =  ^7^^^  ^  "-^"^" 


from    the   result  -^  ^       '  ^ /sv^a^     -     x  2^^ 

we   have,  //^V^^/  ^  ^  A  ^''^^ -^  i:  i  ' A-*  ^^}  ^ -^^  ^ffZ^-^ 


and    substituting   this   value   in    the    second   of   Equations    (168),    we 
have,  after  eliminating    -^  by  its  value,  obtained  from  Equation  (171), 

^.^  =  -^ (172) 

dt    dt  ^  ^      ' 

and    dividing   this  by   the  square    of  Equation   (170), 


dp 

dt  g 

dx  ~         V^  cos^  a 

d~t 


(173) 


MECHANICS    OF    SOLIDS.  143 

but  regarding  z   and  'p  as  functions  of  x,  we  have,  Eq^uation  (171), 

dz 

p  =  -r-  =   -T-' (!'•*) 

■^         dx         dx 

dTt 

and, 

dp 

d  t  dp 
dx  dx 
HT 

whence,  making   V"  =2i/h,  Equation  (173)  becomes 

,  2cs 

dp  € 


d  X  2h.  cos^  a 

and  multiplying  this  by  the  identical  equation, 

d  X  .  ^  1  -\-  p"^  =z  d  s, 
obtained  from  Equation  (174),  we  find, 

2cs 

e  •  ds 


(175) 


-v/  1    +2^^'  dp    =    —    -rr- ^— , 

^  -^        ^  2  h.  cos^  a 

and   integrating, 

p.  ^nrT7  +  log  {p  +  V^+  P')  =  ^-2cAWa^^^^^^ 

in  which  C  is  the  constant  of  integration  ;  to  determine  which,  make 
s  =  0  ;  this  gives  p  =  tan  a  j  and 

C  =  — ; —  +  tan  a  .  J\  +  tan^  a  +  log  (tana  +  -v/l+tan2a)  •  (177) 

2  c  h  cos^  a 

From   Equation  (175)  we  have, 

—  2  c  s 

<?^  =:  —  2  A.  cos^  a  .  e     .      dp  \ 

from   Equation   (171), 

dz  z=  p  .  dx  ; 


144         ELEMENTS     OF     AISTALYTICAL    MECHANICS. 

from  Equation   (n2), 

g  d  f  ■=.  —  dx .  dp  ; 

and  eliminating   the  exponential  factor  by  means  of  Equation   (176), 
we  find, 

c.dx  = ;    .     (ITS) 

p  y/  1  +  'f-  +  log  (i^  +  V'l  +2>')  -  G 

c.dz  =  P^ ^:= ;.     (179) 

p  VTTi^  +  log  {p  +  -/i~+  p^)  -  ^ 

V~cff.dt  =  ~  "^^  ;  .    (180) 

J c  -p  VTT7'  -  log  {p  +  VT+^) 

Of  the  double  sign  due  to  the  radical  of  the  last  equation,  the 
negative  is  taken  because  p^  which  is  the  tangent  of  the  angle  made 
by  any  element  of  the  curve  with  the  axis  of  .-r,  is  a  decreasing 
function  of  the  time  t. 

These  equations  cannot  be  integrated  under  a  finite  form.  But 
the  trajectory  may  be  constructed  by  means  of  auxiliary  curves  of 
which  (17S)  and  (179)  are  the  diflerential  equations.  From  the  first, 
we  have, 

dx  ^  T  .dp; (181) 

and  from  the  second, 

dz  -  T.p.dp; (182) 

in  which, 

T  ^- \ ;  •  (183) 

^      p  -v/l  +i>'  +  log  ( V  +  ^  i  +  V")  -  G 

and  dividing  Equations  (181)  and    (182),  by  dp, 

^  =  ^-' (^"> 


MECHANICS     OF    SOLIDS. 


14:5 


Now,  regarding  x,  ^j,  and  2,  p,  as  the  variable  co-ordinates  of  two 
auxiliary  curves,  T,  and  T .  p,  will  be  the  tangents  of  the  angles 
which  the  elements  of  these  curves  make  with  the  axis  of  p. 

Any  assumed  value  of  p^  being  substituted  in  T,  Equation  (183), 
will  give  the  tangent  of  this  angle,  and  this,  Equation  (184),  multi- 
plied by  dp,  will  give  the  difference  of  distances  of  the  ends  of  the 
corresponding  element  of  the  curve  from  the  axis  of  p.  Beginning 
therefore,  at  the  point  in  which  the  auxiliary  curves  cut  the  axis  of 
2>i  and  adding  these  successive  differences  together,  a  series  of  ordi- 
natcs  X  and  z,  separated  by  intervals  equal  to  dp,  may  be  found,  and 
the  curves  traced  through  their   extremities. 

At  the  point  from 
wliich  the  projectile 
is  thrown,  we  have, 

a;  =  0  ;  2  =  0  ;  ^;=:tan  a, 


and  the  auxiliary 
curves  will  cut  the 
axis  of  p,  in  the  same 
point,  and  at  a  dis- 
tance from  the  origin  equal  to  tan  a.  Let  A  B,  be  the  axis  of  p, 
and  AC,  the  axis  of  x  and  of  2;  take  Aff=  tan  a,  and  let  BzD, 
and  BxE,  be  constructed  as  above. 

Draw  the  axes  Ax  and  Az,  though  the  point  of  departure  A, 
Fig.  (2)  ;  draw  any 
ordinate  c  z,  x.  to  the 
auxiliary  curves.  Fig. 
(I);  lay  off  ^:r,  Fig. 
(2)  equal  to  Cx,  Fig. 
(1),  and  draw  through 
x, ,  the  line  x,  z, 
parallel  to  the  axis 
Az,  and  equal  to  cz, 
Fig.  (1)  ;  the  point 
2^  will    be  a  point    of 

the  trajectory.      The  range  AD,  is  equal  to  ED,  Fig.  (1). 

10 


146  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

By  reference  to  the  value  of  C,  Equation  (177),  it  will  be  seen 
that  the  value  of  T,  Equation  (183),  will  always  be  negative,  and 
that  the  auxiliary  curve  whose  ordinates  give  the  values  of  x,  can, 
therefore,  never  approach  the  axis  of  p.  As  long  as  p  is  positive, 
the  auxiliary  curve  whose  ordinates  are  z,  will  recede  from  the 
axis  p ;  but  when  p  becomes  negative,  as  it  will  to  the  left  of 
the  axis  A  C,  Fig.  (1),  the  tangent  of  the  angle  which  the  element 
of  the  curve  malvcs  with  the  axis  p,  will,  Equation  (ISS)?  become 
positive,  and  this  curve  will  approach  the  axis  p,  and  intersect  it  at 
some   point  as  D. 

Tlie  value  of  p  will  continue  to  increase  indefinitely  to  the  left 
of  the  origin  A,  Fig.  (1),  and  when  it  becomes  exceedingly  great, 
the  logarithmic  term  as  well  as  (7,  and  unity  may  be  neglected  in 
comparison  with  p,  which  will  reduce  Equations  (178)   and  (179)  to 

d.=-^;    d.=  '^; 
c.p^  c.p 

and   integrating, 

a:  =  C"  -  — ;     2  ==  C"  +  -  •  log;?, 
cp  c         ° 

which  will  become,  on  making  p  very  great, 

.r  =  (7':     z  =  C"  +  -\ogp, 
c 

which  shows  that  the  curve  whose  ordinates  are  the  values  of  a;, 
will  ultimately  become  parallel  to  the  axis  p,  while  the  other  has 
no  limit  to  its  retrocession  from  this  axis;  Whence  we  conclude, 
that  the  descending  branch  of  the  trajectory  approaches  more  and 
more  to  a  vertical  direction,  which  it  ultimately  attains ;  and  that 
a  line  G  L,  Fig.  (2),  perpendicular  to  the  axis  ar,  and  at  a  distance 
from  the  point  of  departure  equal  to  C",  will  be  an  asymptote  to 
the  trajectory. 

This  curve  is  not,  like  the  parabolic  trajectory,  symmetrical  in 
reference  to  a  vertical  through  the  highest  point  of  the  curve ; 
the  angles  of  falling  will  exceed  the  corresponding  angles  of  rising, 
the  range  will  be  less  than  double  the  absciss  of  the  highest  point, 
and  the  angle  which  gives    the  greatest  range  will  be  less  than  45°. 


MECHANICS    OF    SOLIDS. 


147 


Denoting   the   velocity   at   any   instant  by   v,   we   have 


df' 


=  (1  +  f^) 


d  x^ 
IF' 


and  replacing    dz-   and   d  t-   by  their  values    in  Equations  (178)  and 
(180),  we  find 

5'-(l  -^P^) 


c 


C  -p  vTTi^  -  log  {p  +  V 1  +  i>2) 


(186) 


and  supposing  p  to  attain  its  greatest  value,  which  supposes  the 
projectile  to  be  moving  on  the  vertical  portion  of  the  trajectory, 
this  equation  reduces,  for  the   reasons   before  stated,  to 


'  =  \/t 


z 


*  Avhich   shows    that    the    final   motion  is  uniform,  and  that  the  velocity 
will    then   be   the    same   as   that   of  a   heavy   body   which  has  fallen 

1  Tc^ 

in  vacuo  through  a  vertical  distance  equal  to =  • 

2  c  '2  (J 

§  152. — When  the  angle  of  projection  is  very  small,  the  projectile 
rises  but  a  short  distance  above  the.  line  of  the  range,  and  the  equation 
of  so  much  of  the  trajec- 
tory as  lies  in  the  imme- 
diate neighborhood  of  this 
line  may  easily  be  found. 
For,  the  angle  of  projec- 
ti#  being  very  small,  p 
will  be  small,  and  its 
second  power  may  be 
neglected  in  comparison 
with  unity,  and  we  may 
take, 

d s  =  dx\    and    s  z=.  x\ 

which  in  Equation  (175),  gives, 

lex 

dp  d'^z  e 

d X    ~    d X"  ~         2h  .  cos^ a 


"y^ 


J? 


(187) 


148  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Integrating, 

2  ex 

ds e ^  ^ 

dx    ~        4c.A.cos2a  ' 

d  z 
making  x  =  0,  we  have  -j—  —  tan  a, 

whence, 

C  =  tan  a  + 


4c  .  h  .  cos^  a 
which  substituted  above,  gives, 

2ci 

dz  el 

=  tan  a  —  ^ — — — r h 


rfa;  '  4:C.h.cos^a       4iC .  h  .cos^a' 


and  integrating  again 


Zcx 


z  =  tan  a.x  —  — - — ; f- 7 5 h  C", 

8  c2  .  A  .  cos2  a       4  c .  A .  cos^  a 

making  a;  =  0,  then  will  z  =  0,  and 

1 


C"  = 


Sc"  .h.  cos^  a 
hence, 

2  =  tan  ax -——i ^    (/"_  2  ca:  -  l)  .     .     (188) 

From  Equation  (172),  we  have, 

g  .di^  =  —  dx  .  dp, 
and  substituting  the  value  of  c?^),  from  Equation  (187), 

CI 

e      .  dx 
dt 


■^  Igh .  cos  a ' 
and  integrating,  making  x  =  0,  when  ^  ==  0, 

(CI  \ 

e    -1)    .     .     .     .     (189) 


c  \/  2gh  .  cos  a 


MECHANICS    OF    SOLIDS.  149 

which   will   give   the   time   of    flight    to   any   point  whose   horizontal 
distance  from   the  piece   is   equal   to   x. 

§  153. — Let  the  projectile  fall  to  the  ground  at  the  point  i),  and 
denote  the  co-ordinates  of  this  point  by  x  =  I,  and  z  =-\  and  sup- 
pose the  time  of  flight  or  i  =  t.  These  values  in  Equations  (188) 
and   (189),  give 

8  c2  .  /i .  cos2  a  (X  +  ^  tan  a)  =  e^''  -  2  c  ?  -  1,  •     (190) 

cos  a.  <r.  c.  y2^A  =  e*^     —1      ....     (191) 

When  the  two  constants  A  and  c,  as  well  as  a  and  X,  are  known, 
these  equations  will  give  the  horizontal  distance  ?,  and  the  time  of 
flight.  Reciprocally,  when  the  quantities  a,  /,  X  and  t  are  known, 
they  give  the  co-efficient  of  resistance  c,  and  the  height  A,  due  to 
the  velocity  of  projection,  and  therefore,  Equation  (135),  the  initial 
velocity  itself. 

Eliminating  the  height  h,  we  fnid 

4\^»*(X  +  Ltana)(e''  -  If  =  g .t-^ .{^'^  _  2c?  -  1);  •  •  (192) 

from  which  the  value  of  c  may  be  found,  and   one  of  the  preceding 
equations  will   give  A,  or   the  initial  velocity. 

It  may  be  worth  while  to  remark  that  if  the  exponential  term 
in  Equation  (188)  be  developed,  and  c  be  made  equal  to  zero,  which 
is  equivalent  to  supposing  the  projectile  in  vacuo,  we  obtain  Equa- 
tion (161). 

LAWS    OF    CENTRAL    FOKCES. 


§  154. — Let  a  body  in  motion 
be  subjected  to  the  action  of  a 
deflecting  force  of  attraction  di- 
rected to  a  fixed  centre.  The 
curve  described  by  the  body  in 
this   case   is   called   an  orbit. 

Assume  the  origin  of  co-ordi- 
nates   at    the  centre,  and   denote 


150 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


the   intensity  of  the  attraction  on   the  unit  of  mass   by  F,  which  we 
will   suppose   to   vary  according  to   any  law.     Then  will 


^         o         y 

cos  a  =z J     cos  p  = 


cos  7  = 5 


in  which  r   denotes  the  radius  vector   of  the  body ;    and  Equations 
(119)  will,  omitting   the  accents,  reduce  to 


y  =  o, 


which  being  integrated,  give 


d?x  d'^z         _ 

d"^  z  d^y  - 


df 


,     .  X -— .  y  —  C  , 

dt  dt    ^  ' 


dx 

11 

dz 
"dt' 


.  z 


y 


dz 
~dT 

dy 
dt 


x=  C'\  \ 
z  =  C" 


(193) 


in  which  C\   C"  and  C",  are  the   constants   of  integration.  * 

Multiplymg  each   by  the  first  power  of  the  variable  which  it  does 
not  contain,  and  adding,  we  have, 

C'z  M-  C"y  +  C"'x  =  0, 

which  is  the  equation  of  an  invariable  plane  passing  through  the 
centre,  and  of  which  the  position  depends  upon  the  constants  C, 
C",  C".  "Whence  we  conclude  that  a  moving  body  deflected  to- 
wards a  centre,  will  describe   a  plane  curve. 

§155. — Take  the  co-ordmate  plane  xy  to  coincide  with  this  plane, 
and   the   Equations  (193)  will   reduce   to 


dy  dx 

— — .  X r—  -y  =  C 

dt  dt     ^ 


(194) 


MECHANICS     OF     SOLIDS.  151 

Substituting  in  Equation  (123)',  MF  for  P;  d  r  for  dp;  making 
P\  P'\  &c.  equal  to  zero,  and  recalling  that  the  angles  a,  /3  and  7 
are   obtuse,  we  have,  ;s?';'/°^  ,  Mr.c^y,ii  ^ywj^j  -„Fy..//ve, 


if  F2 


+  2  J  MFd  r  -  C  =0     •     •     •     •     (195) 


Tliese    two   equations  will   make    known   all    the   circumstances  of 
the  motion. 

g  156. — But    the    discussion    will    be    facilitated    by   transforming 
them    to   polar   co-ordinates ;  and  for   this  purpose  we  have 

a;  =  r  .  cos  a  ;     ?/=?*.  sin  a  ; 
differentiatinc, 

d X  =  dr .  cos  a  —  r  sin  a  (/ a, 
dy  =  d  r  sin  a.  +  r  cos  a.  da. 

Substituting   in  Equation  (194),  we  find 

integrating   again,  we   have, 

frKda  =  C't  +  C", 

and  taking  between  the  limits  i\ ,  a^  and  r^^ ,  a^^ ,  corresponding  to 
the   time  t^  and  t^^, 

f"'''    r'^.daL^  C  {(,,  -  f)       ....     (197) 

But    I  r"^  da   is   double    the  area   described    by  the   motion  of  the 

radius  vector ;  whence  we  see,  Equation  (197),  that  the  areas  de- 
scribed by  the  radius  vector  of  a  body  revolving  about  a  fixed  cen- 
tre, are  proportional  to  the  intervals  of  time  required  to  describe 
them. 


152  ELEMENTS    OF    ANALYTICAL    ilECHANICS. 

Making,  in  Equation  (197),  t^^  —  t^  equal  to  unity,  the  first  mem- 
ber becomes  double  the  area  described  in  a  unit  of  time.  Denoting 
this  by  2  c,  that   equation  gives 

C  =  2c. 

Placing   this   in   Equation   (197),  we  find 


J  r,,  a,, 


r-^.d< 


t.-t,=^-^^^ (198) 

That  is  to  say,  any  interval  of  time  is  equal  to  the  area  de- 
scribed in  that  interval,  divided  by  the  area  described  in  the  unit 
of  time. 

§  157. — The  converse  is  also  true ;  for,  differentiating  Equation  (196), 
we  find, 

Multiplying  by  3f,  and  replacing  M.  -jji    and    M.  -^   by  their 

values   in  Equations  (120),  there  will    result 

Yx  —  Xy  =  0, 

Avhich  is  the  equation  of  the  line  of  direction  of  the  force ;  and  having  no 
independent  term,  this  line  passes  through  the  centre.  Whence  we  con- 
clude, that  a  body  whose  radius  vector  describes  about  any  point 
areas  proportional  to  the  times,  is  acted  upon  by  a  force  of  which 
the  line  of  direction  passes  through  that  point  as  a  centre.  The  force 
will  be  attractive  or  repulsive  according  as  the  orbit  tui-ns  its  con- 
cave or   convex   side   towards   the  centre. 

§158. — Eeplacing  C  by  its  value  2  c,  in  Equation  (196),  and  di- 
viding  by  r^,  we  have  , 

^  =  ^ (199) 

dt  7-2 

The   first    member    being    the    actual   velocity   of  a   point    on   the 


MECHANICS    OF    SOLIDS.  153 

radius  vector  at  the  distance  unity  from  the  centre,  is  called  the 
angular  velociUj  of  the  body.  The  angular  velocity  therefore  varies 
inversely   as    the  square  of  the   radius   vector. 

§  159.— Multiply  Equation  (199)  hy  d  s,  and  it  may  be  put  under 
the  foi'ni, 

ds  2  c 


dt  rda,  ' 

ds 

but 5  is  equal   to   the   sine  of  the   angle  Avhich  the  clement  of 

ds 

the    orbit    makes   with    ^le    radius   vector,    and   denoting  by  p   the 

length  of  the   perpendicular  from    the  centre  on   the   tangent   to   the 

orbit   at   the  place  of  the  body,  Ave  have 

r .  d  oL 
p  =  r.  — — » 
ds 

and 

F=— (200) 

V 

whence,  the  actual  velocity  of  the  body  varies  inversely  as  the  dis- 
tance of  the  tangent  to  the  orbit  at  the  body's  place,  from  the 
centre. 

§  160. — Differentiating  Equation  (195),  we  find, 

VdV  ^  -  Fdr; 

and   taking    the  logarithms  of  both   members  of  Equation  (200), 

log  F  =  log  2  c  —  log  p  ; 

differentiating, 

dV  _        dp 


and  dividing   the   equation   above   by  this, 

!p  •2'''  dp 


V^  =  F.p.'^  =  2F'p.'j-\    .    .    .     (ooi) 


loi  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Whence  we  conclude  that,  the  „ 

velocity  of  a  body  at  any  point 
of  its  orbit  is  the  same  as  that 
which  it  would  have  acquired  had 
it  fallen  freely  fi-om  rest  at  that 
point,  over  the  distance  M E^  equal 
to  one-fourth  of  the  chord  of  cur- 
vature M  G,  through  the  fixed  cen- 
tre— the  force  retaining  unchanged 
its  intensity  at  M. 


Qm 


IGl. — To    find   the   differential   polar    equation   of  the   orbit,  we 


have 


F2  ^ 


dx"  -\-  dy"^  d  >-2  4-  »'^  d  a? 


de 


df 


substituting  this   in   Equation  (195),  differentiating   and    reducing  by 
the   relation 

2r  .dr  .da  =  —  r'^.d^a, 

obtained  by  differentiating  Equation  (196),  we  find 

J-(d^r  -  i^.da^)  -f  i^=  0, 

and    eliminating  dt  by  means  of  Equation  (199),  we   get, 
4  c^    /fZ^  r 


^(3^-'')+^='' ('»=) 


and   making 


/d'^u         \ 


(203) 


From  which  the  equation  of  the  orbit  may  be  found  by  inte- 
gration when  the.  law  of  the  force  is  known ;  or  the  law  of  the 
force   deduced,  when   the   equation   of  the  orbit   is   given. 


MECHANICS     OF     SOLIDS.  155 

In  the  first  case,  the  mtegral  will  contain  three  arbitrary  con- 
stants— two  introduced  in  the  process  of  integration,  and  the  third,  c, 
existing  in  the  diflerential  equation.  These  are  determined  by  the 
initial  or  other  circumstances  of  the  motion,  viz. :  the  body's  velocity, 
its  distance  from  the  centre,  and  direction  of  the  motion  at  a  given 
instant.  The  general  integral  only  determines  the  nature  of  the 
orbit  described :  the  circumstances  of  the  motion  at  any  given  time 
determine   the   species  and   dimensions  of  the    orbit. 

In   the  second  case,  find  the  second  diflerential  co-efficient  of  u  in 

regard   to  a,  from   the   polar   equation  of  the    curve;   substitute   this 

->•  -•--'^-~    -    -c  '*■   ^^r.,,„    ^^y   n^eans   of  the 

^  /      '-     ^   /  '  ^,  m  terms  of  u 

lyUV-   ^.£f^^ ^-^  V- 2-l_^  ^a V  ^ ^ '•  V z^u  V^  from  the  centre; 

.  c^i^  *-  "        intensity  of  the 

^-.^    e  ?-^W<^r-^-,^  V„  ^^W^<^^^r-^.,^..^en  will 

•     Multiplying  by  2  c?  u,   and   integrating, 

^   +  «^  =  C  -  -A_ (204) 

"When   the   radius   vector  is  perpendicular   to   the  orbit,  then  will 

d  r  du        ^ 

cli'zo    .,    — -  =  0  ;     and,  therefore,    -p-  =  0  : 
du.  da. 

and  denoting  the  value  of  the  radius  vector  in  this  position  by   r^ , 
and   the   value  of  the   corresponding   velocity  by   V , ,  we   have 

4c2  -  F^2?-^2. 


loi  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Whence  we  conclude  that,  the 
velocity  of  a  body  at  any  point 
of  its  orbit  is  the  same  as  that 
which  it  would  have  acquired  had 
it  fallen  freely  from  rest  at  that 
point,  over  the  distance  ME,  equal 
to  one-fourth  of  the  chord  of  cur- 
vature M  G,  through  the  fixed  cen- 
tre— the  force  retaining  unchanged 
its  intensity  at  M. 


—X 


161.— To   f    /'J'--^' 


have 


substituting  this 
the  relation 


obtained  by  diffe 


and    eliminating 


and   m  akin  2 


»^-<.^-^z.t.,- 


xj^  iiifaiis  or   x!.quation  (lyj),  we   get, 
{ r*  )  4-  i^  =  0     .     .     .     . 


(202) 


i^  =    4  C2  .  W2 


r 


/d^u  \ 


(203) 


From  which  the  equation  of  the  orbit  may  be  found  by  inte- 
gration when  the.  law  of  the  force  is  known ;  or  the  law  of  the 
force    deduced,  when   the   equation   of  the  orbit   is   given. 


MECHANICS     OF    SOLIDS.  155 

111  the  first  case,  the  mtegral  will  contain  three  arbitrary  con- 
stants— two  introduced  in  the  process  of  integration,  and  the  third,  c, 
existing  in  the  difTerential  equation.  These  are  determined  by  the 
initial  or  other  circumstances  of  the  motion,  viz. :  the  body's  velocity, 
its  distance  from  the  centre,  and  direction  of  the  motion  at  a  given 
instant.  The  general  integral  only  determines  the  nature  of  the 
orbit  described  :  the  circumstances  of  the  motion  at  any  given  time 
determine   the   sj^ecies  and   dimensions  of  the   orbit. 

In  the  second  case,  find  the  second  diflerential  co-efficient  of  u  in 
regard  to  a,  from  the  polar  equation  of  the  curve ;  substitute  this 
ill  the  above  equation,  eliminating  a,  if  it  occur,  by  means  of  the 
relation  between  u  and  a,  and  the  result  will  be  F,  in  terms  of  u 
alone. 

§  1G2. — Let  the  force  vary  directly  as  the  distance  from  the  centre; 
required  the  nature  of  the  orbit.  Denote  by  k  the  intensity  of  the 
force  on   a   unit  of  mass   at   the   unit's   distance ;   then  will 

F=kr  =  —-, 
u 

and   this,  in   Equation  (203),  gives, 

d^u    ,  k 


da^  4  c- 1/-^ 

•     Multiplying  by  2  d  ic,   and   integrating, 

d  a^  4  c'^  u^ 

"When    the   radius   vector  is  perpendicular   to   the  orbit,  then  will 

,  <^  '■        rv  11/.  du        ^ 

c/t:o    ..    -.—  =  0;     and,  therefore,    -—  =  0  : 
ft  a  da. 

and  denoting  the  value  of  the  radius  vector  in  this  position  by   r^ , 
and   the   value  of  the   corresponding   velocity  by   V^ ,  we   have 

4c2  -  F/?-,2. 


loG  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and   the  value   of  C,   will   be   given   by 

1  k 

which,  substituted  above,   gives 


d^  ~        r/-  F,2  F,2  r,2  m2 

whence, 

1  "Hu  .du 

da  =  -- 


adding  and   subtracting   under  the   radical   the   expression, 

F,a  +  r^k 

the   above   may   be   written, 

2r2F2 


1                 F,2  —  r/  ^ 
da,  z=  —-  


2u  .d  u 


and   integrating, 


2  z.\  2 


/  /2  r^2  p^^2.  ^2    _    7^2    _   ;.^2  ^.X 

Y   1  -  V~  F/  -  7-/  A;  / 


2  (a  +  ?)  =  sm       • p^^2  _  ,.^2  yt ' 

in  which  9  is  the   constant  of  integration. 

Let  the  axis  from  which  a  is  estimated,  coincide  with  the  normal 
radius  vector ;  then,  when 

a  =  0,    will    «2  =  —2 ; 
^/ 

and  we   have, 

.  -1  * 

2  9  =  sm      1  =  —; 

which   substituted   above,  gives, 

.   /  -rrX  .  -1  2r  2  V,Ku^  -  F/  -  r.-'k 


MECHANICS    OF    SOLIDS.  157 

and   from  which  we  have, 

sin»  (l«  +  -^)  =  cos  2«  =  _^_^___^__^  ; 

replacmg    cos  2  a    by   cos2  a  —  siii2  a,    finding   the  value   of    v^,   and 
substituting   therefor  — )  we   obtain,  after  a  slight  reduction, 

=       ....    (205) 


U^  -  i   y.-^-^.^:^ 


^  -z.  ^  /-^  t^,  •^_  .  r 


./*ccr^ 


— j.,*./!^.     2-  -i^eiCtt^ 


/'y,''-^i  £  j^ 7 V-^zVi--* -^-i? 


-    g 


le 

n 

le 

11 

f 

attraction  at  the  unit's  distance.      The  result  of  this  proposition  is  of 

the  greatest  importance  in  physical  science,  as  we  shall  have  occasion 

to  see  when  we  come  to  the  subjects  of  Acoustics,  Optics,  &c. 

§  1C4. — Let  the  central  force  vary  inversely  as  the  square  of 
the   distance :    required   the   orbit. 

Employing  the  same  notation  as  in  the  last  proposition,  we  shall 
have 


156  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

and   the  value   of  (7,   will   be   given   bj 

(7-1+  A, 

^    —     ,.  2    ^      F  2' 

which,  substituted  above,   gives 

h  h 

d  a?  r^  V  '^  V ^  r^  u^ 

whence 


d  tt2          F,2  4-  r/  k 

^' 

da?   ~        r^  V;^ 

F,2  r^  u^ 

^'-c^ 

addir 


the 


-2Lz: 


/ 


and 


"T"    _ _  Jy  «5t^^' 


in  wl  7^-   7^  -7^^//^       _    _2^.:^    fzp-i) 

radius  vector ;  then,  when 

a  =  0,    will    tt^  -—  — j^^ . 


and  we   have. 


2  9  =  sm      1  =r  —  ; 


2 
which   substituted    above,  gives, 

•1    2r/  F,2.„2  _   F^2  _,.^2X; 

F2   -  T?k  ' 


^  ^  +  -^^  =  sin 


MECHANICS    OF    SOLIDS.  157 

and   from  which  \vc  have, 

sin^  i^+  -)  =cos2u=  vr-r;h 


replacmg    cos  2  a    by   cos^  a  —  sin^  a,    finding   the  value   of   i/^,    and 


substituting   therefor  —1  we   obtain,  after  a  slight  reduction, 


— -  cos^  a  +   tT^  •  sin^  a 


(205) 


which  is  the  equation  of  an  ellipse  referred  to  the  centre  as  a  pole, 
the  semi-axes   being 

r,  and  — r^' 
/I' 

§  163. — The  time  required  to  describe  the  entire  orbit  being 
denoted  by  T,  we  have,  Equation  (198), 

*  •  ^/  •  -7^ 
T  = ^  =  — (206) 

2 

Whence  we  conclude,  that  the  orbit  described  by  a  body  under  the 
action  of  a  central  force  which  varies  directly  as  the  distance  from 
the  centre,  is  an  ellipse ;  and  that  the  time  required  to  perfoi-m  one 
entire  revolution  about  the  centre,  is  constant,  being  the  same  for  all 
orbits,  great  and  small,  and  is  dependent  solely  upon  the  intensity  of 
attraction  at  the  unit's  distance.  The  result  of  this  proposition  is  of 
the  greatest  importance  in  physical  science,  as  we  shall  have  occasion 
to  see  when  we  come  to  the  subjects  of  Acoustics,  Optics,  d:c. 

§  164. — Let  the  central  force  vary  inversely  as  the  square  of 
the   distance :    required   the   orbit. 

Employing  the  same  notation  as  in  the  last  proposition,  we  shall 
have 


158         ELEMENTS    OF    ANALYTICAL    MECHANICS, 
which  in  Equation  (203)  gives 


d  a?  4  c^  ' 


multiplying  by  2  (?m,  and  integrating 


To  determine  the  constant  C,  we  -p. 

recall   that 


du       d  \r) 
d  a~      da 


dr 


r^,  d  a,        r.  tan  s 


in  which   s  denotes  the   angle  made 

by  the  radius  vector  with  the  element  of  the  curve;  and  if  this  be 
known  for  any  radius  vector  r^,  corresponding  to  the  place  from 
which  the  body  is  projected,  then  will  s  be  the  angle  of  projection 
in   reference   to   the   centre,   and, 


C  = 


1 


r/ .  tan^  s        ? 
but,  Equation  (200), 


+  \~ 


2Jc 


1 


2  k 


4  c^  fj       r^  .  sin"  e       4  c^  r^ ' 


whence, 


/•/  .  sin^  £         4  c'^         4  c^  r^ 


4c'^  r. 


in  which    F,  is   the   velocity   when   the   radius  vector  is   7\.     Substi- 
tuting this   above,   we   get 


dti 
~d^* 


V,\r:^-2Jc    .       F 


4  c^  r. 


+ 


16  c»* 


("-4^) 


MECHANICS     OF    SOLIDS.  159 

whence 

I  —  du 

da  =. 


and  integrating 


a  +  (p  =  cos 


V        4c2r,  10  c»' 


a  =  0,  when 
acing  u  by  its 


.  .  (207) 

f  9) 

being   at    the 
learest    vertex. 


r  = 


■.y,.,c,^  1  +  e.cos(a  +  (p) 


we    find  r/r"^'''*'?  A (??"•«  -y- 


■''"P" 


TX  ^  !-t-~^(K>y>^ 


(208) 


T.  k-" 


but  4  c2  =  r,2  .  r;^  .  sin2  e,  whence 

■£Jl~.,^  i-:.  e  =  ^-'  V  ^  '  ■  r,  ■  F,'  sin'  .  +  1  ;     •     •     •      (200) 

f-o'-.'Ju  e // ,yi  <i  t  ,   --5-.^   j'y  }^  e  'y  /-<'  vf   .. 

and  /a;' //,<>/,'.,/>  A^^/„    /•/  ,'9  •"■  J'"''^'"'' .. 

„(l_,3)=i£=!VIifmL^     .     .     .     (2,0) 


158         ELEMENTS    OF    ANALYTICAL    MECHANICS, 
which  in  Equation  (203)  gives 


multiplying  by  2  t?  m,  and  integrating 


To  determine  the  constant  C,  we  -„• 

'  T\                   \. — -^-. 

recall   that  '  ,  ,               ^ 

r  - 


du 

■u 


d^  d"-  ^  —  fe^     .        ,   -4//<5<..^**.^— ^     ^=^ 


^^^..  .;.j/'^ili-_^7r  '"^^''"^ 


in  which   s  de  .^^>.       (^^^s^;/   ^^^    aL^  /^^  ^sfeyi*..!^.^,^ 

bv  the  radius  ,^-3— — — :; ,  „ 

known    for    an  /       ^_^^-z_       ^'      •■                                            '            \^ 

which   the  bod  ^^t-«^v     <j-t^>-c^  wvt^     -^-p-^^/cii.   (tJ J  .t^^-c^   ^zair') - 

in   reference   t  A=  ^   ••      2^^^-^   „    Ai^'- =  1^^^,^^^-^'^^       ^> 


but,  Equation  ( 


Avhence, 


1  F2 


r/  ,  sni2  £        4  c^         4  c^  ?• 


4  (.2  y. 


in  which    F,  is  the   A^elocity   when   the   radius  vector  is   r^.     Substi- 
tuting this   above,   we   get 


c?  a*  4  c2  r,  '    16  c»*      V        4  c 


"^  16^~  \        4c2^/ 


MECHANICS    OF    SOLIDS.  159 

whence 


(fa  = 


—  (I  u 


V       4  c2  r.        ^  10  c*  L  ^  4  c-/ 


4c2r,       ''"lOc«* 


and  integrating 


•  -1  ^  -  4^ 

a  +  (p  =  COS 


The    value    of   9   is    found    by    the    condition  that   a  =  0,   when 
M  =  J- .      Taking    the   cosine   of    both   members,   replacing  u  by  its 

value  —5  and   reducing,  we   have 


r 

4c2 


Ar  +  v/-'— ^ — '^— ; ^^ h  ^    •  cos  (a  +  <p) 


(207) 


which  is  the  equation  of  a  conic  section,  the  pole  being  at  the 
focus,  and  the  angle  (a  +  9)  estimated  from  the  nearest  vertex. 
Comparmg   it  with   the   equation, 

cA..rr,;  A.G..,^^  _  a{\-  e^)  ,       . 

^  ,.y,.  .c,^      ^  -  1  +  e  .  cos  (a  +  (p)  ^     ^ 

we    findr/("7-rA'r  A(??"»  y-  r. ,       /, 

_  (T?r,-2^-).4c2  . 

but  4  c^  =  r^  .  V^  .  sin^  s,  whence 

-;.--/^^^-e2  =  i^^Il^.r,.F,2sin^s  +  l;  •  •  •  (209) 
and  j^oj'Me  A  ■./•.'- ^„/..    //  ?c  >,.  s,v,  „f  ,. 

At  A/ 


160         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Multiplying  both  numerator  and  denominator  of  the  first  factor 
in  the  second  member  of  Equation  (209),  by  M  r^,  the  orbit  will  be 
an  ellipse,  parabola  or  hyperbola,  according  as 

,^^.,        2Mk  t 

■^'^      *  I       ^  2  '  /  ' 

I 

that  is  to  say,  according  as  the  living  force  of  the  body,  at  any 
point  of  its  orbit,  is  less  than,  equal  to,  or  greater  than  twice  the 
quantity  of  work  its  weight  at  that  point,  supposed  constant,  w^ould 
generate  were  it  to  fall  freely  through  the  corresponding  radius 
vector   to   the   centre. 

And  it  is  a  remarkable  fact,  that  the  s])ecies  of  conic  section  is 
wholly  independent  of  the    direction  in  which   the  body  is  projected. 

In  the  case  of  the  ellipse  and  hyperbola,  the  major  or  transverse 

axis   is 

^kr, 


F  2  r.  -2  k' 


(211) 


which  is   also  independent   of  the    direction  of  the   projection. 

In   the   case   of  the   parabola,  the    distance  i),   from   the   focus   to 
vertex,  is   given   by  the  equation 

The   position    of   the    transverse   axis    in    reference    to   the    radius 
vector  1\ ,  is   obtained   by   making  a  =  0,  and  r  =  r^;   thus,    /'« ^  >^) 

a{\  —  e"^)  1  V^ .  r  .  sin^  s  —  k 

cos  <p  =  — ^ =  -, 

r^e  e  ke 

Making   a  +  9  =  90°,  the    corresponding  value  of  r  will  give  the 
semi-parameter ;  that  is, 

4  (.2         y  2  _  y-i  ^  sin2  g 


r  = 


k  k 


(211)' 


MECIIAXICS     OF    SOLIDS.  161 

and  because  the  semi-conjugate  axis  is  a  mean  proportional  between 
the  semi-parameter  and  semi-transverse  axis,  we  have,  denoting  the 
semi-conjugate   axis  by  b, 

a 


b  =  r^.V^.sms.  aJ— (212) 

which  depends  upon   the   angle   of  projection. 

§  165. — To  give  a^  example  of  the  reverse  process,  let  it  be 
required  to  find  the  law  of  the  force  which  will  cause  a  body  to 
describe   a   conic  section  when   directed   to   one   of  the  foci. 

The   equation  of  the   orbit   referred    to    the   focus,  is 

a  (1  -  e2) 

r  ^  — ^ — —  • 

1  +  e  cos  a  ' 

whence, 

1  1+6  cos  a 

r  a  [1  —  e^) 

and, 

d^  u         —  e  cos  a 
d  a^         a  (1  —  e^) 

which,  substituted   in  Equation  (203),  give 

ET     '-A    ->    .  f  —  e  cosa         1  4-  e  cos  a\ 
\a  (1  —  e^)         a  (I  —  e^)y 

reducing  and  replacing   u  by  its  value   — 5  we  have, 

r 

4c2  1 

F  =  --j ^  .  — (213) 

a(l  —  e^)     7-2  ^       ^ 

and  from   which   we   conclude,  that   the   only   law  is   that   of  the   in- 
verse  square   of  the   distance. 

§  166. — If  e  be   made   equal   to   zero,  the   conic  section   becomes  a 
circle,  in   which  case   a  =z  r,  and   the   above   becomes 

~     a3 
11 


162  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Also  in   Equations    (199)  and  (200),  we   have 
r  =.  a,    and   ^  =  a  ; 

whence, 

da        2c       ,    ^^       2c 
-—  =  --  and   V  =  — ; 
dt         a^  a  ' 

that  is  to  say,  both  the  angular  and  absolute  velocity  will  be  con- 
stant. ^ 

Denoting   the    time   required    to    perform    an    entire   revolution   by 
2" — called    the  ^)eriOcZ/c   thne.     Then,  Equation  (198),  will 

^        ■T^a-  2*  a  ,^,  ,\ 

^  -  —  =  -F" ^^^^^ 

§  167. — Resuming  Equations  (120),  we   have 

dx 

df-  dt 

and  performing  the  operation  indicated,  regarding  the  arc  of  the 
orbit  as  the  independent  variable,  we  have,  after  dividing  both  nu- 
merator  and   denominator  by  ds"^, 


X  =  M- 


dt      d^x         dx     cP-t 
d  s       d  6-2  d  s     d  s^ 

J? 


fds"^      d"fx     dx     d  S'^       d^  t~\ 
^  ^    ■  Ufi  '  'dW~  rfV  dJ'  '  d^i ' 


d  s^      d"^  t                d"S  ds 

^''"''^  '     'dfi  '  ~d¥  ^  ~  ITc- '  IT  ~       ' 

whence, 

x=j/.rF^.^'.+  4^.^i. 

L          d «?«  ds       dfi  A 
In  like   manner, 

-^^  ~        L       rf^V^  ds     df'  J' 


MECHAXICS    OF    SOLIDS.  163 

Squaring   and   adding, 

^   O    T72        d?S    {<1^       (Pz  d>/        ip7J  dz        (Pz\ 

d  fi   ^d  s      d  s-         d  s      d  s~         d  s      d  s~^ 

+   \dp  +  772  +  ^/    Kdj)   '^^  ' 

but,  denoting  the  radius  of  cui'vature  by  p,  we  have 

^d^  xy     /d^  yy     APzy  _  _i_ 
\d^)  +  \d^)  ^  \de)  ~  f  ' 

and    multiplying   the    second    term    of    the    second    member   of   the 
preceding   equation   by    —5  it   may   be  put   under   the  form, 

M  V^     M .  d"  s  ^d  X        d"  X     ^    d  y        d"^  y    ^     d  z         d"^  z' 


d  f^ 


/«  X        d-  X         d  y        d^  y         d  z         d^  z\ 
Vc/  s     "  d  s^         d  s         d  s'^  d  s  d  5-/ 


in  which  5  denotes  the  angle  made   by  the  element  of  the  curve  and 
radius   of  curvature ;    also 

d  X"         d  y-         d  z^ 

J^  '^  JV^   '^  d^   "^       ' 

whence,  substituting  for  A'-  +  V^  +  Z^  its  value  i?^,  we  hnve 

P„       J/2  F*  MV^    Il.d^s  .    ^  ...     /d-^sy 


and   comparing    this  with    Equation  (5G)  we  find  that   H   is    equal   to 
the   resultant   of  the   two  component  forces 

and  M  •        -  > 

p  d  t- 


164  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

which  make  with  each  other  the  angle  S.  But  8  is  equal  to  90°, 
and   therefore 

^-^+--ey (-) 

The  second  of  these  components  is,  Equation  (13),  the  inten- 
sity of  the  reaction  of  inertia  in  the  direction  of  the  tangent,  and  the 
first  is  therefore  its  reaction  in  the  direction  of  the  radius  of 
curvature. 

This  first  component  is  called  the  centrifugal  force,  and  may  be 
defined  to  be  the  resistance  which  the  inertia  of  a  body  in  motion 
opposes  to  whatever  deflects  it  from  its  rectilinear  path.  It  is  measured, 
Equation  (215),  by  the  living  force  of  the  body  divided  by  the  radius 
of  curvature.  The  direction  of  its  action  is  from  the  centre  of 
curvature,  and  it  thus  differs  from  the  force  which  acts  towards  a 
centre,  and  which  is  called  centripetal  force.  The  two  are  called 
central  forces. 

§  168. — If  the  component  in  the  direction  of  the  orbit  be  zero,  then 
will 

and  denoting  the  centrifugal  force  by  F, ,  we  have 

F,  =  ^^ (216) 

and  integrating  the  next  to  the  last  equation,  we  have 

"5-    =   F=  C; 
d  t 

in  which  C  is  the  constant  of  integration.  Whence,  the  velocity  will 
be  constant,  and  we  conclude  that  a  body  in  motion  and  acted  upon 
by  a  force  whose  direction  is  always  normal  to  the  path  described, 
will  preserve  its  velocity  unchanged. 


MECHANICS    OF    SOLIDS.  165 


EOTAKY   M0TI02f. 

§  169. — Having  discussed  the  motion  of  translation  of  a  single 
body,  we  now  come  to  its  motion  of  rotation.  To  find  the  circum- 
stances of  a  body's  rotary  motion,  it  will  be  convenient  to  transform 
Equations  (118)  from  rectangular  to  polar  co-ordinates.  But  before 
doing  this,  let  us  premise  that  the  angular  velocity  of  a  body  is  the 
rate  of  its  rotation  about  a  centre.  The  angular  velocity  is  measured 
by  the  absolute  velocity  of  a  point  at  the  unites  distance  from 
the  centre,  and  taken  in  such  position  as  to  make  that  velocity  a 
maximum. 

§  170. — Both  members  of  Equations  (38)  being  divided  by  d  t, 
give 


dx' 

dt 

=  z' 

d-\. 
'  dt 

-  y' 

d  9 
dt 

dy' 
dt 

=  x' 

d  <p 
'    dt 

-  z' 

d  zi 

'  IT 

dz' 
dt 

=  y' 

d  zi 
'    dt 

-  x'  . 

d-\. 
dt 

(217) 


in  which  the  first  members  taken  in  order,  are  the  velocities  of  any 
element,  as  m,  in  the  direction  of  the  axes  x,  y,  z,  respectively,  in 
reference  to  the  centre  of  inertia,  §  75,  while 

dzi       d-\/       dcp 
dt         d  t       d  t 

are  the   angular  velocities   about   the   same  axes  respectively. 

Denoting  the  first  of  these  by  v^,  the  second  by  v^,,  and  the  third 
by  Vj,  we  have 


dvi  c?4/  d  (^ 

'dJ  ^  "''   TT  ""  '" '    'dT 


=  V.  ;      .      .      .      .      (218) 


166  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

and  Equations  (217)  may  be  written 


dx' 
dt 

dy^ 
dt 

dz' 


—  z'.v^  —  y'.v,^" 


■=.  -X    .V,  —  S   .  V, 


y  .  v^  —  X  .  v^ 


(219) 


§171. — If  an  element  m  be  so  situated  that  its  velocity  shall  be 
equal  and  parallel  to  that  of  the  centre  of  inertia,  then,  for  this 
element,  will  each  of  the  first  members  of  Equations  (219)  reduce 
to  zero,  and 

g'     V     —   w'     V     —    0        "1 


X    .  V, 


z'  .v,  =  0, 
y' .  V,  —  x' .  Vy  =  0  ; 


(220) 


the  last  being  but  a  consequence  of  the  two  others,  these  equations 
are  those  of  a  right  line  passing  through  the  centre  of  inertia, 
every  point  of  which  will  have  a  simple  motion  of  translation 
parallel  and  equal  to  that  of  the  centre  of  inertia.  The  whole 
body  must,  for  the  instant,  rotate  about  this  line,  and  it  is,  there- 
fore, called   the  Axis  of  Instantaneous  Rotation. 

§  172. — Denote  by  a^ , 
/3^ ,  7^ ,  the  angles  which 
this  axis  makes  with  the 
CO  ordinate  axes  or,  y,  2, 
respectively.  Then,  tak- 
ing any  point  on  the  in- 
stantaneous  axis,  Avill, 


^x'^  +  y'2  +  s'^' 


cos/3^  = 


y.r'2   +  y'2  ^   ,'2 


cos  y^  =. 


y^'2    +   y'2   +    .J2 


MECHAXIC3    OF    SOLIDS, 
and  eliminating  x\  y'  and  z\  by  Equations  (220), 


167 


cos/3^ 


•/v/  +  V    2  +  v^ 


V^x    +  %^  +  ^z 


[221" 


/'2'';5^'7//-  ^^'^^'V^r^^  rr.v/^"^*-,-; 

; £^^XjL 

iiviiv  i*u  a.  uuib  0  vAiotauv/c  ixuui  uic  ctjiitre  01  niei'titi  J 


(222) 


and  making 

x'  cos  a^  +  y'  cos  /3^  +  z'  cos  7^  =:  0,      •     • 

the  point  takes  the  position,  giving  the  maximum  velocity.  In  this 
case  V  becomes  the  angular  velocity,  and  we  have,  denoting  the 
latter  by  v^. , 


166  ELEMENTS     OF     AN-ALYTICAL    MECHANICS. 

and  Equations  (217)  may  be  written 
dx'        , 

dy' 

dz' 


(219) 


§171.— If  an    element  m  be  so   situated  that   its  velocity  shall  be 


stantaneous  axis,  Avill, 


K 


cos/S^  = 


cosy^  = 


-v/^+3/''  + 


ya;'2  +  y'2  _j_  ^.2 


MECHANICS     OF    SOLIDS, 
and  eliminating  a-',  y'  and  z\  by  Equations  (220), 


167 


cos  a. 


^V/+V2  +  V^2 


COS  /3^  = 


-/v/  +  V   '-^  +  V^2 


X' 


COS  7^  = 


VV/-  +  V   2  +  V,2 


(22i: 


which  will  give    the    position   of    the   instantaneous   axis    as   soon  as 
the   angular  velocities   about   the  axes  are    kno^\^l. 

§  173.— Squaring    each   of   Equations    (219),  taking   their   sum   and 
extracting   square  root,  we  find 


/ 


(/x'2  +  dy'-''  +  rfs'2 


df^ 


=:t;  =  -/(3'.v,^-y'.vJ2+(a:'.v,-0'.vj2+(y'.v^-a;'.v^)2: 


Eeplacing  v^. ,  v    and  v,  by  their  values  ^obtained  by  simply  clearing 
the   fractions  in    Equations    (221),    this   becomes 


V  =  -/v/  +  v^2  _^  v^2  X  ^j.'2  _^  y '2  _|.  .'2  _  ^^.'  cos  a^  +  y'  cos  (3^  +  z'  cos  y^y, 

which  is  the  velocity  of  any    clement   in   reference   to   the  centre  of 
inertia. 
^Making 

a:'2   +   y'2   +   ^'2   ^    1^ 

we  have  the  element  at  a  unit's  distance  from  the  centre  of  inertia ; 
and  makins 


;'  cos  a^  +  y'  cos  /3^  +  z'  cos  y^  =  0, 


(222) 


the  point  takes  the  position,  giving  the  maximum  velocity.  In  this 
case  V  becomes  the  angular  velocity,  and  we  have,  denoting  the 
latter  by  v . , 


V,.  =  Vv"T^7T^ 


(223) 


168  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Equation  (222)  is  that  of  a  plane  passing  through  the  centre 
of  inertia,  and  perpendicular  to  the  instantaneous  axis.  The  position 
of  the  co-ordinate  axes  being  arbitrary,  Equation  (223)  shows  that 
the  sum  of  the  squares  of  the  angular  velocities  about  the  three 
co-ordinate  axes  is  a  constant  quantity,  and  equal  to  the  square  of 
the  angular  velocity  about   the   instantaneous   axis. 

§  174.— Multiply  Equation  (223),  by  the  first  of  Equations  (221), 
and  there  will  result 

v^  .  cos  a^  =  v^ (224) 

whence  the  angular  Velocity  about  any  axis  oblique  to  the  instanta- 
neous axis,  is  equal  to  the  angular  velocity  of  the  body  multiplied 
by  the  cosine  of  the  inclination  of  the  two  axes. 

§175. — Equation  (223)  gives  v.,  when  v^,v^,v^,  are  known.  To 
find  these,  resume  Equations  (118),  and  write  for  the  moments  of  the 
extraneous  forces  in  reference  to  the  axes  a;,'  y,'  z,'  through  the  centre 
of  inertia,  JV^,  M^,  Z,,  respectively,  then  will 

difterentiating  the  first  of  Equations  (219),  with  respect    to  i,  we  find 

dfi   ~''y'  dt  ^'  dt  dt     ^         dt      ^ 

d  z'  d  v' 

and  replacing  -^  and  ---■,  by    their    values    given    in    the    second 

and  third  of  Equations    (219), 


dfi 


=  -(V  +  ^0-^'  +  ^-%-^'  +  ^-'-''  +  ^/*''"c77^"^' 


MECHANICS     OF    SOLIDS. 


169 


in  the  same  way 


-_  =  _  (^v^-  4-  v^2j  .  y    +  v^  .  v^  .  a:    +  v^  .  v^  .  z'  +   — '  .  a;' 
and  these  values  in  the  first  of  Equations,  ('225),  give 


dt 


dt 


/d^y  d-x       \ 

\dt"  dt^    -'/ 


l.  =  i,.(226) 


Similar  equations  will  result  from  the  remaining  two  of  Equations 
(225)  ;  then  by  elimination  and  integration,  we  might  proceed  to  find 
the    values    of  v^,    v     and    v.,    but    the    process    would    be    long   and 

tedious.  It  will  be  greatly  simplified,  however,  if  the  co-ordinate 
axes  be  so  chosen  as  to  make  at  the  instant  corresponding  to  t, 


-Lmx'  y'  =  Q;    2  m  z'y'  =  0;    i:  7n  z'  x'  =  0  ; 


(227) 


which   is    always   possible,    as    will    be    shown   presently.      This  will 
reduce  Equation  (22G)  to 

dv 

—■  '  2  m  (y'2  _^  .^'2)  _|_  v^  .  Vj, .  2  7?i  (y'2  —  x'~)  =  L,  • 

The   other    two     equations    which    refer   to   the   motion    about   the 
axes  y'  and  x',  may  be  written  from   this   one.     They  are, 

dv„ 


d 

dv 


-^  .  2  m  (j;'2  +  2'2)  +  v^  .  V.  .  2  in  (x'2  —  z'2)  =  M^ 


—-.2m  (y'2  +  ^'2)  +  v^  .  v_  2  m  (y'2  _  ,'2) 


=  N^. 


ITO 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


The  axes  x\  y',  z',  which  satisfy  the  conditions  expressed  in 
Equations  (227),  are  called  the  principal  axes  of  Jigure  of  the  body. 
And  if  we  make 


2  m  ,  (y'2  +  ^'2)  ^  ^^ 
2  m  .  (a;'2  +  z'^)  =  B, 
2m.(/2+  ,'2^^  C; 

we  find,  by  subtracting  the  second  from   the   third, 
2  m  .  (y'2  -  a;'2)  =  C  -  J3, 

the  third  from   the    first, 

2  m  .  (.r'2  -  2'2)  =  A  -  C, 

and    the    second   from  the    first, 

2  m  .  (y'2  _  2'2)  =A  -  B; 

which    substituted   above,   give, 


(227)' 


(228) 


By  means  of  these  equations,  the  angular  velocities  v^ ,  v^ ,  v^ ,  must 
be  found  by  the    operations  of  elimination   and  integration. 

I  176. — It  is  i^lain  that  the  quantities  A,  B  and  C,  are  constant 
for  the  same  body  ;  the  first  being  the  sum  of  the  products  arising 
from  multiplying  each  elementary  mass  into  the  square  of  its  dis- 
tance from  the  principal  axis  s',  the  second  the  same  for  the  prin- 
cipal axis  y',  and  the  third  for  the  principal  axis  x'.  The  sum 
of  the  products  of  the  elementary  masses  into  the  square  of  their 
distances  from    any  axis,  is  called  the  moment  of  inertia  of  the  body 


MECHANICS     OF    SOLIDS. 


m 


in  reference  to   this  axis.     A,  B  and   C  are  called  principal  moments 
of  inertia. 

g  177. — To  show  that  in  every  body  there  is  a  system  of  rectan- 
gular co-ordinate  axes,  and  in  general  only  one  system  which  will 
satisfy  the  conditions  expressed  by  Equations  (227),  assume  the  for- 
mulas'^ for  the  transformation  from  one  system  of  rectangular  co- 
ordinates  to    another    also  rectangular.     These  are 


'  =  a:,  cos  {x'  x)  +  y  cos  {x'  y)  -\-  z  cos  (x'z),l 
'  =  X  cos  {y' x)  +  2/.  cos  {y' y)  +  s .  cos  {y'  z),  )■ 
'  =  a;  cos  {z' x)  4-  y  cos  {z' y)   +  z.cos  (z's),J 


(229) 


in  which  {x'  x),  {y' x)  and  {z' x),  denote  the  angles  which  the  new 
axes  x',  y',  z\  make  with  the  primitive  axis  of  x ;  [x' y),  {y' y) 
and  (2'y),  the  angles  which  the  same  axes  make  with  the  primitive 
axis  of  y,  and  {x'  s),  (y'  s)  and  (2'  z),  the  angles  they  make  with  the 
axis  z. 

Assume  the  common 
origin  as  the  centre  of  a 
sphere  of  which  the  radius 
is  unity  ;  and  conceive  the 
points  in  which  the  two 
sets  of  axes  pierce  its  sur- 
face to  be  joined  by  the 
arcs  of  great  circles ;  also 
let  these  points  be  con- 
nected with  the  point  iV, 
in   which   the    intersection  -^^  ^    v  ?  ^  <?  /« w  X  ^^< 

of    the    planes   xy   and   x' y'   pierces    the   spherical  surface  nearest  to 
that  in  which  the  positive  axis  x  pierces  the    same.     Also,  let 
6  =:  Z'  A  Z  —  X'  jyX,  being  the  inclination  of  the  plane  x'  y'  to  that 

of  0"  y. 
^|/  =  XA  X  being    the    angular   distance    of    the    intersection   of    the 

planes  x  y  and   x'  y',  from  the    axis   x. 
(p  =  N A  X'  being    the    angular    distance    of   the    same    intersection 

from  the    axis  x' . 


172  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Then,  in    the    sj^herical    triangle  X'  NX, 

cos  {x'  x)  =  cos  -^^ .  cos  9  +  sin  ■\y .  sin  9  .  cos  &  ; 

In  the  triangle  Y'  NX,  the  side  N  Y'  =  —  +9,  and 

cos  (y'  x)  =  —  cos  -v]^  .  sin  9  +  sin  4^ .  cos  9  .  cos  ^  ; 

In   the   triangle  Z'  NX,  the  side  NZ'  =  ~,   and 

cos  (2'  x)  =  sin  4-  •  sin  6, 

And   in   the   same   way  it  will   be  found  that 

cos  [x'  y)  =  —  sin  4^  •  cos  9  +  cos  -^  .  sin  9 .  cos  6  ; 
cos  [y'  ?/)  =  sin  -^  .  sin  9  +  cos  -^  .  cos  9  .  cos  6  ; 
cos  (2'  y)  =  cos  4^ .  sin  6  ; 
cos  (x'  2)  =  —  sin  9  .  sin  6  ; 
cos  {y'  2)  z=  —  cos  9  .  sin  &  ; 
cos  {z'  z)  =  cos  6  ; 

and   by  substitution   in   Equations  (229), 

x'  =  X  (sin  4^ .  sin  9  .  cos  6  +  cos  4^  •  cos  9) 

+  y  (cos  4* .  sin  9  .  cos  6  —  sin  ■]^ .  fijfA  (p)  —  z  sin  9  .  sin  ^, 
y'  =  X  (sin  4^ .  cos  9  .  cos  6  —  cos  4^  •  sin  9) 

+  y  (cos  4-  •  cos  9  .  cos  ^  +  sin  4> .  sin  (p)  —  z  cos  9  .  sin  ^, 
z'  =  X  sin  4^  •  sin  6  +  y  cos  4>  •  sin  ^  +  2  cos  6  ; 

or    making,  for  salve  of  abbreviation, 

J)  z=  X  cos  -^  —  y  sin  4', 

^  =  X  sin  4^  •  cos  &  +  ?/  cos  4^  •  cos  ^  —  z  sin  6, 

the  above   reduce   to 

x'  =  E .  sin  9  +  Z) .  cos  9, 
y'  =  E .  cos  cp  —  D  .  sin  9, 
z'  =  X  .  sin  4^ .  sin  ^  +  y  .  cos  4'  •  sin  ^  +  2  .  cos  6. 


MECHANICS     OF    SOLIDS.  173 

Substituting   these  values  in   the    equations 

Hm.x' .-)/  =  0;     2m..r'.2'  =  0;     i:m  .  7/ .  s'  =  0  ; 

we  obtain  from  the  first, 

sin  (p .  cos  <p .  2  7?i  (^2  _  _j)2)  -j-  (cos2 9  —  sm^  jp)  I.mE.D  =  0, 

or,  replacing  sin  (p.cos<p,  and  cos- 9  —  sin^ip,  by  their  equals  ^  sin  2  9, 
and  cos  2  9,  respectively, 

sin  2(p  .  2  m  (^2  _  2)2)  +  2  cos  29 . 2  7?ii) .  ^  =  0 ;  .  •  •  (230) 


and  from    tHe\.second,/andj  third,(  respectively, 

cos  9  .  2  77i .  ^ .  s'  —  sin  9  .  2  ??i  Z> .  ;r'  =  0,  •     •     •     (231) 
sin  (p  .  1  m  .  £ .  i^  +  cos  cp  .  1 771  D  .  z'  —  0.  •     •     •     (232) 

Squaring  the  last  two  and  adding,  we  find 

(2  wi .  U.  z'y  +  {H  7)1.1).  z'Y  =  0. 

which  can  only  be  satisfied  by  mailing 


.U.z'  =  0-) 
.D.z'  =  0.)' 


Im.E.z 
2  m 


(233) 


These  equations  are  independent  of  the  angle  9,  and  will  give  the 
values  of  4^  and  & ;  and  these  being  known.  Equation  (230)  will  give 
the  angle  9. 

Replacing  E  and  D  by  their  values,  we   have 

^.  s'  =  sin  ^  .  cos  &  {x^  sin^  4^  +  2  .r  y  sin  ■1'  cos  ■j'  +  3/^  cos^  4^  —  z"^) 
+  (cos2  ^  —  sin2  (3)  (x  ^  sin  -j/  +  ij  z  .  cos  ■\,)  , 

D  .z'  =  sin  &  \x  y  (cos2  4.  —  sin^  4.)  +  (.c2  —  y"^)  sin  4.  cos  4.} 
+  cos  ^  (a:  s  cos  4^  —  yz  sin  4^) . 
and  assuming 

2  m  a;2   z=  A' ;  I.  m  7/-  —  B' ;  ^Z  m  z"-  =  C ; 
l7nxy  =  E'  \  ^mxz  =  F' ;  'Zmyz  =z  U\ 

and  replacing  sin  6  .  cos  6,  and  cos^  &  —  s\n^  6,  by  their  respective 
values,  ^  sin  2  ^,  and  cos  2  6,  Equations  (233)  become 

sin2^(^'sin2  4.  +  2  ^' sin  4/ cos  4.  -f  i?'cos2  4.  —  C)  , 

—  V  5 


,+  2  cos  2  ^  {F'  sin  4.  -{-  H'  cos  4.) 


ni 


174:    ELEMENTS  OF  ANALYTICAL  MECHANICS, 


sill  &{E' .  (cos2  ^  -  sin2  \)  +  (^'  _  ^') .  sin  4.  cos  4.} 
4-  cos  d  {F'  cos  ■\,  —  H'  sin  4.) 


j.., 


in  which    A',  B',   C\  £J',  F'  and   H',  are   constants,   depending   only 
upon  the    shape  of  the    body  and    the   position  of  the  assumed   axes 

X,  y,  2- 

Dividing    the    first    by    cos  2  ^,  and   the    second  by    cos  ^,    they 
become 

tan  2  ^ .  {A'  sin^  4.  +  2  ^'  sin  4.  cos  4.  +  ^'  cos  2  4.  _  (/)  )  _ 

+  2  (i^  sin  ^  +  H'  cos  4.)  j  ^    '  ^^     ^ 

tan  &  .  {E'  (cos2  4,  —  sin^  4.)  +  {A'  -  B')  sin  4.  cos  4.}  ) 

+  Fcos^  -  H'  sin  -^  \^^'  ^^^^^ 

From  the  first  of  these  we  may   find  tan   2  ^,  and  from  the    second, 
tan  ^,  in  terms  of  sin  4^,  and  cos  4- ;  and  these  values  in  the  equation 

tan2^=-^i^, (236) 

1  —  tan2  ^  '^       '' 

will  give  an  equation  from  which  4^  may  be  found. 
In  order  to  effect  this  elimination  more  easily,  make 

tan  4^  =  w, 
whence 

•     1  "  1  1 

sni  4^  =  — 1^=  5  cos  4^  = 


making  these  substitutions  above,  we  find 


.a„2.=  -  2(^'„  +  /f')/r  +  l?- 


A'  ifi  +  2  E'  u  -\r  B'  —  C'(\  +  tt^y 


(F'  -  H'  u)-x/\  +  V? 
tan  ^  =  —      ^  '  ^ 


E'  (1  -  tt2)  4-  (yr  -  B')  u 

which  in  Equation  (236)  give 

f        B'  F'-F'C'-E'H'    )  1 
^^^'-^^  +  ^^'-''>'\  +  (C'H'-A'H'+ET)„\    (  =  0.  .  (337) 
+  {F'u  +  H') .  {F'  -  H'  ti-Y     j 
or, 


MECHANICS     OF     SOLIDS.  175 

■which  is  an  equation  of  the  third  degree,  and  must  have  at  least 
one  real  root,  and,  therefore,  give  one  real  vjilue  for  4-.  This  value 
being  substituted  in  either  of  the  preceding  equations,  must  give  a 
real  value  for  d,  and  this  with  vj/,  in  either  of  the  Equations  (231) 
or  (232),  a  real  value  for  9  ;  whence  we  conclude,  that  it  is  always 
possible  to  assume  the  axes  so  as  to  satisfy  the  required  conditions 
and  that  there  arc  in  every  body  at  least  one  system  of  j^rincipal 
a.ves,  at  right  angles  to  each  other. 

The  three  roots  of  this  cubic  equation  arc  necessarilv  real-  and 
they  represent  the  tangents  of  the  angles  which  the  axis  x  makes 
with  the  lines  in  which  the  three  co-ordinate  planes  x'  y',  y'z',  x' z'  cut 
that  of  a;y;  for  there  is  no  reason  why  we  should  consider  one 
of  these  angles  as  given  by  the  equation  rather  than  the  others,  and 
the  equations  of  condition  arc  satisfied  when  we  interchange  the 
axes  x'  y'  z'.  Hence,  in  general,  there  exists  only  one  set  of  prin- 
cipal axes.  If  there  were  more,  the  degree  of  the  equation  would 
be  higher,  and  would,  from  what  we  have  just  said,  give  three  times 
as   many  real   roots  as  there  are  systems. 

If  E'  =  //'  =  i^'  =  0,  Equation  (237)  will  become  identical ;  the 
problem  will  be  indeterminate,  have  an  infinite  number  of  solutions, 
and  the  body  consequently  an  infinite  number  of  sets  of  principal 
axes.     Such  is  obviously  the   case  Avith   the  sphere,  spheroid,  &c. 

MOMENT   OF   INEKTIA,    CENTRE   AND   EADIUS    OF   GYRATION. 

§  178. — The  quantities  A,  B  and  C,  in  Equations  (227)'  are  the 
moments  of  inertia  of  the  body  in  reference  to  the  principal  axes. 
To  find  these  moments  in  reference  to  any  other  axes  having  the 
same  origin  as  the  principal  axes,  denote  by 

x',  y',  s',  the  co-ordinates  of  m  referred   to    the   principal  axes  ;  by 
a-,  y,  2,  the    co-ordinates    of    the     same    clement   referred    to    any 
other   rectangular  system  having  the  same  origin  ;  and  bv 
C",  the   moment  of  inertia  referred  to  the   axis  z ; 
then  from  the  definition, 

C  =  2  m .  (.1-2  +  y2-)  _  V  „,  ^2  _^  2  „j  ^2  . 


1Y6  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

but   by    the    usual  formulas  for   transformation, 

ar*  =  ax'  +  h  rf  +  c  z\ 
y  =  a'x'  +  b'y'  +  c' z\ 
z  =  a"x'  +  b"y'  +  c"z\ 

in  which  a,  b,  &c.,  denote  the  cosines  of  the  angles  which  the  axes  of 
the  same  name  as  the  co-ordinates  into  which  they  are  respectively 
multiplied  make  with  the  axis  corresponding  to  the  variable  in  the 
first  member. 

Substituting  the  values  of  x  and  y  in  that  of  C,  and  reducing  by 
the   relations, 

2  7nx'y'  =  0',     2  ??i  x' z'  z=  0 ',     I.  m  y' z'  =:  0  ; 

and  we  have, 

C  =  a"2  .  2  m  {f  +  P)  +  i"2  .  2  ??i  {z^^  +  i"^)  +  c"^ .  2  m  (i^  +  f)  ; 

and  by  substituting  A,  B  and   C  for  their  values,  this  reduces  to 

C  =  a"2  A  +  6"2  ^  -f  c"2  (7     .     .     .     .      (238) 

That  is  to  say,  the  moment  of  inertia  with  reference  to  any  axis 
passing  through  the  common  point  of  intersection  of  the  principal 
axes,  is  equal  to  the  sum  of  the  products  obtained  by  multiplying 
the  moment  of  inertia  with  reference  to  each  of  the  principal  axes, 
by  the  square  of  the  cosine  of  the  angle  which  the  axis  in  question 
makes  with  these  axes. 

I  179. — Let  A,  be  the  greatest,  and  (7,  the  least  of  the  moments 
of  inertia,  with  reference  to  the  principal  axes;  then,  substituting  fur 
a"2,  its  value,  1  —  i"^  —  c"2,  in  Equation  (238),  we  have 

C  =^  A  -  6"2  i^A  -  B)  -  c"2  {A  -  C).      •     •      (239) 

By  hypothesis,  A  —  B,  and  A  —  C,  are  positive  ;  therefore,  C  is 
always  less  than  A,  whatever  be  the  value  of  b'\  and  c". 

Again,  substituting  for  c"2  its  value  1  —a"^  —  h""^  in  Equation 
(238),  we  get 

C"  =:  C  +  a"2  (^  -  C)  +  6"2  {B  -  C)      .     .     •     (240) 
and  C  must  always  be  greater  than   C. 


MECHANICS     OF     SOLIDS.  177 

Whence,  we  conclude  that  the  principal  axes  give  the  greatest  and 
least  moments  of  inertia  in  reference  to  axes  through  the  same  point. 
If  A  be  equal  to  B,  then  will  Equation  (239),  become 

C"  r=  (1  -  c"2)  A  +  c"2  C, (241) 

and  this  only  depending  upon  c",  we  conclude  that  the  moment  of 
inertia  will  be  the  same  for  all  axes  making  equal  angles  with  the 
principal  axis,  z'.  The  moments  of  inertia,  with  reference  to  all  axes 
in  tHe  plane  x'  y\  are,  therefore,  equal  to  one  another.  But  all  the 
axes   in   the   plane   x'  y\  which   are  at   right   angles   to   one   another, 


2  m  (.r^  +  ?/2) 

which  is  the  moment  of  inertia  with  reference  to  any  axis,  z,  parallel 
to  the   axis  z\  through  the  centre  of  inertia,  we  have 

2  m  {x--  +  ,f)  :=  2  m  [  (.r,  +  x'f  +  (y,  +  y'f] 

+  2x^  .S-mx'  +  2y,  .  2  7/iy'; 

but  from  the   principle  of  the  centre  of   inertia, 

Imx'  =  0,    and     1.my'  =  0; 

whence,  denoting   by  d   the  distance   between   the   axes  z  and  z\  and 
by  M  the  whole   mass, 

2  m .  (.c2  +  y2)  _  2  ?«  (x'2  +  y'2)  +  MiP  ■     ■     ■     (242) 
12 


176  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

but  by   the   usual  formulas  for   transformation, 

X  =  ax'  +  ^  y'  +  c  z', 
y  =  a'x'  -{-  b'y'  ^  c' z\ 
z  =  a"x'  +  h"y'  +  c"z\ 

in  which  a,  5,  &c.,  denote  the  cosines  of  the  angles  which  the  axes  of 
the  same  name  as  the  co-ordinates  into  which  they  are  respectively 
multiplied  make  with  the  axis  corresponding  to  the  variable  in  the 
first  member. 

Substituting  the  values  of  x  and  y  in   that  of  C",  and  reducing  by 


Dy  tne  square  ot  the  cosme  ot  the  angle  which    the    axis   in    question 
makes  with  these  axes. 

S  179, — Let  A^  be  the  greatest,  and  (7,  the  least  of  the  moments 
of  inertia,  with  reference  to  the  principal  axes ;  then,  substituting  for 
a"2,  its  value,  1  —  h"'^  —  c"2,  in  Equation  (238),  we  have 

C  =  A-  b"^  {A  -  B)  -  c"2  [A  -  C).      •     •      (239) 

By  hypothesis,  A  ~  B,  and  A  —  C,  are  positive ;  therefore,  C  is 
always  less  than  A^  whatever  be  the  value  of  b'\  and  c". 

Again,  substituting  for  c"^  its  value  1  —a"^  —  b"^  in  Equation 
(238),  we  get 

C"  =  (7  +  «"2  (^1  -  C)  +  6"2  {B-  C)      '     '     '     (240) 
and  C  must  always  be  greater  than  C. 


MECHANICS    OF    SOLIDS.  177 

Whence,  we  conclude  that  the  principal  axes  give  the  greatest  and 
least  moments  of  inertia  in  reference  to  axes  through  the  same  point. 
If  A  be  equal  to  B,  then  will  Equation  (239),  become 

C  =  {I  -  c"2)  A  +  c"2  C, (241) 

and  this  only  depending  upon  c",  we  conclude  that  the  moment  of 
inertia  will  be  the  same  for  all  axes  making  equal  angles  with  the 
principal  axis,  z'.  The  moments  of  inertia,  with  reference  to  all  axes 
in  tVte  plane  x'  y',  are,  therefore,  equal  to  one  another.  But  all  the 
axes  in  the  plane  x'  y\  Avliich  are  at  right  angles  to  one  another, 
are,  §  175,  when  taken  with  z',  principal  axes,  and  we,  therefore, 
conclude  that  the  body  has  arf  indefinite  number  of  sets  of  principal 
axes. 

If,   at   the   same   time,  we   have  A  =  B  =  C,  then  will   Equation 
(238)  reduce  to 

C  =  C  =  A  =  B. 

that  is,  the  moments  of  inertia  are  all  equal  to  one  another,  and  all 
axes  are  principal,  the  Equation,  (238)  being  satisfied  independently 
of  a",  b",  c". 

§  180. — Resuming    Equations,    (33),   and    substituting    the    values 
of  a",  y,  2,  in  the  general  expression, 

2  m  (a:2  +  y2) 

which  is  the  moment  of  inertia  with  reference  to  any  axis,  2,  parallel 
to  the   axis  2',  through  the  centre  of  inertia,  we  have 

2  m  {x"-  +  y-)  =  2  m  [  {x^  +  x'f  +  (y,  +  y'f] 

=  2  m  (x'2  +  y'^)  +  (.r/-  +  y;-) .  2  m 
+  ^x^  .-Lmx'  +  2y,  .  2/«y'; 

but  from  the   principle  of  the  centre  of   inertia, 

2  m  x'  —  0,    and     2  ?«  y'  —  0  ; 

whence,  denoting  by  d  the  distance  between  the  axes  z  and  z\  and 
by  M  the  whole   mass, 

2  m .  (.c2  +  y2)  ^  2  771  (x'2  +  y'2)  4-  McP  ■     .     ■     (242) 
12 


1T8  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

That  is,  the  moment  of  inertia  of  any  body  in  reference  to  a  given 
axis  is  equal  to  the  moment  of  inertia  with  reference  to  a  parallel 
axis  through  the  centre  of  inertia,  increased  by  the  product  of  the 
whole  mass  into  the  square  of  the  distance  of  the  given  axis  from 
that   centre. 

And  we  conclude  that  the  least  of  all  the  moments  of  inertia  is 
that  taken  with  reference  to  a  principal  axis  through  the  centre  of 
inertia. 

I  181. — Denote  by  r  the  distance  of  the  elementary  mass  m  from 
the  axis  z,  then  will 

j,2    -_    ^2    _|_  ■y2^ 

and 

Now,  denoting  the  whole  mass   by  M,  and  assuming 
2  m  7-2  =  Mk^^ 


we   have 


=V^ (-) 


The  distance  h  is  called  the  radius  of  gyration,  and  it  obviously 
measures  the  distance  from  the  axis  to  that  point  into  which  if  the 
whole  mass  were  concentrated  the  moment  of  inertia  would  not  be 
altered.  The  point  into  which  this  concentration  might  take  place 
and  satisfy  the  condition  above,  is  called  the  ceiitre  of  gyration. 
When  the  axis  passes  through  the  centre  of  inertia,  the  radius  h 
and  the  point  of  concentration  are  called  principal  radius  and  prin- 
cipal centre  of  gyration. 

The  least  radius  of  gyration  is,  Equation  (243),  that  relating  to 
the  principal  axis  with  reference  to  which  the  moment  of  inertia  is 
the  least. 

If  k,  denote  a  principal  radius  of  gyration,  we  may  replace 
2  m  (.c'2  J-  y'2)  in  Equation  (242)  by  Mk;^,  and  we  shall  have 

2«ir2  =  J/F  =  i/(A-/  +  cZ2)      ....     (244) 


MECHANICS    OF    SOLIDS. 


179 


If  the  linear  dimensions  of  the  body  be  very  small  as  compared 
with  d,  we   may  write  the   moment  of  inertia  equal   to  M<P. 

The  letter  h  with  the  subscript  accent,  will  denote  a  principal 
radius  of  gyration. 

§  182. — The  determination  of  the  moments  of  inertia  and  radii 
of  gyration  of  geometrical  figures,  is  purely  an  operation  of  the  cal- 
culus. Such  bodies  are  supposed  to  be  continuous  throughout,  and 
of  uniform  density.  Hence,  we  may  write  d  J/  for  w,  and  the  sign 
of  integration  for  2,  and   the  formula  becomes 


2  mr"^ 


=  / 


dM.r'^ 


(245) 


Example  1. — A  j^hysical  line  about  an   axis    through   its   centre  and 
perpendicular   to   its    length. 

Denote   the  whole   length  by  2  a ;   then  ' 

2a:dr:'.M:dM, 


whence, 


and 


dr 

dM  =  M'—, 

2a 


/a  ^2 

M-  —  -dr  = 
■1         2  a 


y  3 

If  the   axis  be   at  a   distance  d   from   the   centre,  and   parallel  to 
that   above,  then.  Equation  (244), 


k  =  y/\a^  +  d"^. 

Example  2. — A    circular   plate    of    uniform   density   and    thickness, 
about  an  axis   through   its   centre   and  perpendicular   to  its  plane. 


180  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Denote  the  radius  by  a;  the  angle  XA  Q  \Y 

by  6  ;   the   distance  of  dM  from    the   centre 
by  r;   then, 


ira^ir.dd.dr  :  :  M:  dM: 


whence, 


dM=  M- 


r .d  r . d& 


and 


pa     pV  7.3      y/  J. 


dr  = 


Ma^ 


If  a- 


k.  =  a  -JJ, 


2M.- 
0  a'- 

1:1 
and  for  an  axis   parallel  to   the  above  at  the  distance  c?, 

Exam-pie  3. — The  same  body  about  an  axis  through   its   centre  and 
in   its  2^l(t^^^' 


As  before, 


dM  =  M- 


r.dr.dd 


in  which  r  denotes  the  distance  of  d  M  from  the  centre ;  and  taking 
the  axis  to  be  that  from  which  6  is  estimated,  the  distance  of  the 
elementary  mass  from   the   axis  will   be   r  sin  6,  and 

'  Jo   Jo  qf  a?  ^lll  a'^ '^  0    ll  O 


M    ['"■  a^ 

JlfA-2  =  —  /    rKdr  =  M -, 


and 


Tc,=\a, 
and  about   an   axis   parallel  to  the   above  and   at  the  distance  d. 


h 


4 


MECHANICS    OF     SOLIDS. 


181 


It  is  obvious  that  both  the  axes  first  considered  in  Examples  2 
and  3  are  principal  axes,  as  are  also  all  others  in  the  plane  of 
the  plate  and  through  the  centre,  and  if  it  were  required  to  find 
the  moment  of  inertia  of  the  plate  about  an  axis  through  the  centre 
and  inclined  to  its  surface  under  an  angle  9,  the  answer  would  be 
given  by  the  Equation  (238), 

Mk^"  =  -T  Ma^  sin^  <p  +  i"  -^«^  cos^  9 
=  ijfa2(l  +  sin^ifj), 

and  for  a  parallel  axis  whose   distance  is  d, 

MP  =  J/  (  T  a^  (1  +  sin2  9)  +  d^^  • 

Example  4. — A  solid  of  revolution  about  any  axis  perpendicular  to 
the   axis   of  the   solid. 

Let  D  A'  E  be  the  given  axis, 
cutting  that  of  the  solid  in  A'-  Let 
A'  be  the  origin  of  co-ordinates, 
P  M  =  y;  A'  F  z=  x;  AA'=m; 
A'  B  =  w,  and  V  =  volume  of  the 
solid. 

The  volume  of  the  elementary 
section  at  P  will  be 


and 
whence, 


•^f  y"^ .  dx, 

V  :  M  :  :  -rr  .y^  .d  X  :  dM\ 

d  M  =^  —  •  -x  •  y^  '  d  X, 


m 

A 

A 

"M 

i\ 

['j 

• 

J 
F 

v 

and  its  moment  of  inertia  about  MM',  is,  Example  3, 

■y-'^-y-'dx.-. 

and  about  the  parallel  axis,  D  JE, 
M 


V 


.'!(.y''-.dx{\y'^  +  a;2) 


182  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

therefore, 


But 


whence. 


m     V 
V  =z    /     "K  xP-  dx'. 

jm  (4  ^*  "^  x^  y'^).dx 


k^  = 


The  equation   of  the   generating  curve  being   given,  y  may  be  elimi- 
nated and  the  integration  performed. 

Example   5. — A  sphere   about   a   line    tangent   to  its   surface. 
The  equation  of  the  generatrix  is 

2/2  —  2  a  a;  —  x"^; 

in  which  a  is  the  radius  of  the  sphere.     Substituting   the  value  of  y"^ 
in  the  last  equation,  recollecting  that   m  =  0,   and  w  =  2  a,  we  have 

f  °(a2  x^  +  ax^  —  ^x*)dx 
Jo  7     „ 

7.2 =  -  a^- 

^     —  nia  5 

/      (2  a  a;  —  a;^)  c?  a; 

Also  Equation  (244), 

A:  2  =  ^-2  _  a2  =  I  a2 

and 


Jc,  ^  a,. ,    ^ 
5 


Centrifugal  force  arising  from  the  rotation  of  the  earth  upon  its  axis. 
§  (182)'.— If    Fi    denote    the   angular  velocity  of  a  body  about   a 
centre,  then  will   F-  p  V„  and  Equation  (216)  becomes 

F,  =  31  V,' p. 


MECHANICS     OF    SOLIDS. 


183 


The  earth  revolves  about  its  axis 
A  A'  once  in  twenty  four  hours, 
and  the  circumferences  of  the  par- 
allels of  latitude  have  velocities 
\\hich  diminish  from  the  equator  to 
the  poles.  To  lind  the  law  of  this 
diminution,  let  W  be  the  weight  of 
a  body  on  the  surface  of  the  earth 
in  any  parallel  of  which  B',  is  the 
radius  ;    its  centrifugal  force  will   be 


W 

9 


•V,^E'; 


in    which     W  is    the    weight    of    the    body,    and    V^,  is    the    angular 

W 


velocitv  of  the    earth.       Substituting  M  for  — ,  we  have 

F,  z:^  M  F,^  R'. 


Denoting  the  equatorial  radius  C  E  —  C  P,  by  i?,  ar-d  the  angle 
C  P  C  =  P  C  E,  which  is  the  latitude  of  the  place,  by  9,  we  have 
in  the  ti-iangle  P  G  C", 

R'  =z  R  cos  (p  ; 

which  substituted  for  R'  above,  gives 

F^  =  M  T7  i?  cos  <p (-245)' 

The  only  variable  quantity  in  this  expression,  when  the  same 
mass  is  taken  from  one  latitude  to  another,  is  9 ;  whence  tve 
conclude  that    the    centrifugal  force  varies  as  the  cosine  of  the  latitude. 

The  centrifugal  force  is  exerted  in  the  direction  of  the  radius  R' 
of  the  parallel  of  latitude,  and 
therefore  in  a  direction  oblique 
to  the  horizon  T  T'.  Lay  off 
on  the  prolongation  of  this 
radius,  the  distance  P  If,  to 
represent  this  force,  and  resolve 
it  into  two  components  PiV 
and  P  T,  the  one  normal,  the 
other  tangent  to    the   surface  of 


184  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

,► 

the  earth ;  the  first  will  diminish  the  weight  W  by  its  entire  value, 
being  directly  opposed  to  the  force  of  gravity,  the  second  will  tend 
to   urge   the   body  towards   the  equator. 

The  angle  HPN  is  equal  to  the  angle  P  C  JS,  which  is  the 
latitude,  denoted   by  9  ;   whence    the   normal    component 

PJSf  =  PH  X  cos  cp  =  F^ .  cos  cp  =  M  V{  R  cos2 9, 

and 

P  T  =  P  H  sin  p  =  F,.  sin  (p  —  MV'^  R.  sin  9  cos  9  ; 
but, 

sin  9  .  cos  9  =  i  sin  2  9  ; 

therefore, 

PT  =z\MV'{Rsm^(^ ; 

whence  we  conclude,  that  the  diminution  of  the  weights  of  bodies 
arising  from  the  centrifugal  force  at  the  earth's  surface,  varies  as  the 
square  of  the  cositie  of  the  latitude ;  and  that  all  bodies  are,  in  con- 
sequence of  the  centrifugal  force,  urged  towards  the  equator  by  a  force 
which    varies   as    the    sine    of  twice    the    latitude. 

At  the  equator  and  poles  this  latter  force  is  zero,  and  at  the 
latitude  of  45°  it  is  a  maximum,  and  equal  to  half  of  the  entire 
centrifugal  force   at   the   equator. 

At  the  equator  the  diminution  of  the  force  of  gravity  is  a 
maximum,  and  equal  to  the  entire  centrifugal  force ;  at  the  poles 
it  is  zero.  The  earth  is  not  perfectly  spherical,  and  all  observations 
agree  in  demonstrating  that  it  is  protuberant  at  the  equator  and 
flattened  at  the  poles,  the  difference  between  the  equatorial  and 
polar  diameters  being  about  twenty-six  English  miles.  If  we  sup- 
pose the  earth  to  have  been  at  one  time  in  a  state  of  fluidity,  or 
even  approaching  to  it,  its  present  figure  is  readily  accounted  for  by 
the  foregoing  considerations. 

The  force  of  gravity  which  varies,  according  to  the  Newtonian 
hypothesis,  directly  as  the  mass  and  inversely  as  the  square  of  the 
distance  from   the   centre  of  the    earth,  is,  therefore,  on   account  of  a 


MECHAXICS    OF     SOLIDS.  1S5 

difference  of  distance  and  of  the  centrifugal  force  of  the  earth  com- 
bined, less  at  the   equator   than   at   the   poles. 

To  find  the  value  of  the  centrifugal  force  at  the  equator,  make, 
in  Equation  (245)',  M  =.  I  and  cos  9  =  1,  which  is  equivalent  to 
supposing   a   unit  of  mass   on   the   equator,  and  we   have 

F,=  V-R, 

in  which   if  the   known   radius    of   the  equator  and   angular   velocity  /*' 
be  substituted,  we   shall  find 

F^  ^  F^i2  =  0,  1112. 

By  the  aid  of  this  value,  it  is  very  easy  to  find  the  angular 
velocity  with  which  the  earth  should '  rotate,  to  make  the  centrifugal 
force  of  a   body  at  the    equator  equal  to  its  weight. 

By  the  new  rate  of  motion, 

^r  =  32  ,  1937  =  F,'2  R  ■ 

f 
in   wliich    32  ,  1937  is  the  force  of  gravity  at   the   equator. 

Dividing   the    second   by  the  first,  and  we  find 

32,1937  r/2       ^^^ 

-OTTm    -1^5=289,  nearly; 

whence, 

F/  =  17  r, ; 

that  is  to  say,  if  the  earth  were  to  revolve  seventeen  times  as  fast 
as  it  does,  bodies  Avould  possess  no  weight  at  the  equator ;  and  the 
weights  of  bodies  at  the  various  latitudes  from  the  equator  to  the 
poles  diminishing  in  the  ratio  of  the  squares  of  the  cosines  of  lati- 
tude, the  weights  of  all  bodies,  except  at  the  poles,  would  be  allbcted. 


EMPULSIVE   FORCES. 

§  183. — We  have  thus  far  only  been  concerned  with  forces  whose 
action  may  be  likened  to,  and  indeed  represented  by,  the  pressure 
arising  from   the  weight   of  some    definite   body,  as   a    cubic   foot   of 


186  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

distilled  water  at  a  standard  temperature.  Such  forces  are  called 
incessant^  because  they  extend  their  action  through  a  definite  and 
measurable  portion  of  time,  A  single  and  instantaneous  effort  of 
such  a  force,  called  its  intensity,  is  assumed  to  be  measured  by  the 
whole  effect  which  its  incessant  repetition  for  a  unit  of  time  can 
produce  upon  a  given  body.  The  effect  here  referred' to  is  called 
the  quantity  of  motion,  being  the  product  of  the  mass  into  the 
velocity  generated.     That  is.  Equations  (12)  and  (13), 

P  =  M.r,  =  M'^=M';i;    .    .    .     .(246) 

in  which    F^,  denotes  the  velocity  generated  in  a  unit    of  time. 

The  force  P,  acting  for  one,  two,  or  more  units  of  time,  or  for 
any  fractional  portion  of  a  unit  of  time,  may  communicate  any  other 
velocity  F,  and  a  quantity  of  motion  measured  by  M  V.  And  if 
the  body  which  has  thus  received  its  motion  gradually,  impinge  upon 
another  which  is  free  to  move,  experience  tells  us  that  it  n\ay 
suddenly  transfer  the  whole  of  its  motion  to  the  latter  by  what 
seems  to  be  a  single  blow,  and  although  we  know  that  this  transfer 
can  only  take  place  by  a  series  of  successive  actions  and  reactions 
between  the  molecular  springs  of  the  bodies,  so  to  speak,  and  the 
inertia  of  their  different  elements,  yet  the  whole  eflect  is  produced  in 
a  time  so  short  as  to  elude  the  senses,  and  we  are,  therefore,  apt  to 
assume,  though  erroneously,  that  the  effect  is  instantaneous.  Such 
an  assumption  implies  that  a  definite  velocity  can  be  generated  in  an 
indefinitely  short  time,  and -that  the  measure  of  the  force's  intensity 
is,'  Equation  (246),  infinite. 

In  all  such  cases,  to  avoid  this  difficulty,  it  is  agreed  to  take  the 
actual  motion  generated  by  these  blows  during  the  entire  period 
of  their  action,  as  the  measure  of  their  intensity.  Thus,  denoting 
the  mass  impinged  upon  by  M,  and  the  actual  velocity  generated 
in   it   when   perfectly   free   by   F,   we   have 

P  =  MV.  =  M.  ^J, (247) 

in   which   P,    denotes  the    intensity   of    the    force's    action,    and    the 
second  member  of  the  equation  the  resistances  of  the  body's  inertia.    " 


MECHANICS     OF    SOLIDS. 


187 


Forces  which  act   in  the   manner  just   described,   by   a   blow,    are 
called   impulsive  forces. 

MOTION    OF   A   BODY   UNDER   THE   ACTION   OF   IMPULSIVE   FORCES. 

g  184. — The  components  of  the  inertia  in  the  direction  of  the  axes 

Xi/z,  are  respectively 

,^dsdx        ,,    ifx 
JV/-  — •— =  J/-  — ; 
a  t  as  at 


a  t   a  s  at 


ds   dy 
d  t   ds 

ds   dz 


M-  :!-".-:  =  M-  — ; 

dtds  d  t 

which,    substituted   for    the    corresponding    components    of    inertia   in 
Equations  (^1)  and  (5),  give 

dx    ^ 


2  P  cos  a  =  2  7?i  -        , 
d  t 

dv 
1  F  cos  13=  Urn-  --^; 
d  t 


2  F  cos  y  =r  2  ??i 


dz 
dTi 


(248) 


/  ,    d?/         ,    dx\     ^ 
2  F {x'  cos (3  —  y'  cos  a)  =  2  m  \x'  •  ^^  — V  •  ^t)  ' 

/  ,     dx         ,    dz\ 
2  F  (z'  cos  a  —  x'  cos  7)  =  2  m  \^z'  '  ^  —  ^   '  J-J  ' 

2  P(/ cosy  -2' cos  ^)  =  ^m{^y'-~-z'-^J- 


(•249) 


In  which  it  will  be  recollected  that  x  y  z  arc  the  co-ordinates  of  m, 
referred  to  the  fixed  origin,  and  x'  y'  2',  those  of  the  same  mass 
referred  to  the  centre  of  inertia. 


MOTION    OF   THE   CENTRE     OF    INERTIA. 
I 

§185. — Substituting    in    Equations    (248),    for    dx,    d y,    d  z,    their 
values   obtained   from  Equations  (34),  and  reducing  by  the   relations 

^mdx'=:0;   lmdy'  =  0;   Zmdz'  =  0;    •     •     .  (250) 


188         ELEMENTS     OF     ANALYTICAL    MECHANICS, 
given  by  the  principle  of  the  centre  of  inertia,  we  find 

Q,  X 

2  P  cos  a  =  — ^  .2m; 
d  t 

2P 008/3=-^. 2m;    )■ 
2  P COS  7  —  —j-lm; 
and  substituting  31  for  2  w,  we   have 


2  P  cos  a  =  M 


2  P  cos  jS  =  jlf 


d  x^ . 

■  dt ' 


2  P  cos  7  =  3/  •  -r-^ ; 


(251) 


which  are  wholly  independent  of  the  relative  positions  of  the  elements 
of  the  body,  and  from  which  we  conclude  that  the  motion  of  the 
centre  of  inertia  will  be  the  same  as  though  the  mass  were  concen- 
trated in    it,  and  the   forces   applied  immediately  to  that  point. 

§  186. — Replacing  the  first  members  of  the  above  equations  by 
their  values  given  in  Equations  (41),  and  denoting  by  V  the  velocity 
which  the  resultant  Rj  can  impress  upon  the  whole  mass,  then  will 

2P  cosa  =  ifFcos  a;    2P  cos6  =  J/ Fcos  i  ;    2  P  cose  =  ilf  Fcosc; 

substituting   these   above,  we  find 

d  X, 


V.  cos  a'  = 

V .  cos  h  = 
V .  cos  c  = 


dt 

dt 
dz^ 

~dr 


(252) 


MECHANICS    OF    SOLIDS. 


189 


and  integrating, 

x^  =  V-cosa.t  +  C", 

y^  =  V.cosb.i  +  C", (253) 

z,  =  V.  cos  c.i  +  C",  , 

and   eliminating   t   from   these   equations,    V  will   also  disappear,   and 
we  find, 


C  cos  c  —  C"  cos  a     '\ 


cos  a 


I. 


(254) 


*.-fi^'<^ 


.^i„^~^^  <i^  c-Tc-A/  -^        }   consequence 
\  ^.^/.l..M4:  ^^       e.      This   line 

y  V"  >i'ces. 

^^    ^;;*^.>w-"^  ^a&'^'^^s^   ^^  ^^^^  ^-  "3^^7^  es,  will    move 
JLAe±^   . -^  jf^U.     o^  -«—   -^-^^^  (^'^^)  ..  -esultant. 


MOTION   ABOUT   THE   CENTKE   OF   INERTIA. 

§187. — Substituting,  in  Equations  (249),  for  dx,  dy  and  o?^,  their 
values   from    Equations    (34),    reducing   by 

Imx'  —  0, 
2  m  y'  =  0, 
2  m  z'  —  0, 
and  we  find, 

2  P  {x'  cos  (3  —  y'  cos  ot)  =  2  m  (^.c'  •   yj   —  y'  -ryj  ; 


2  P  (s'  cos  a  —  .r'  cos  /)  =  2  m  yz'  ■ x'  •  -r-J  ; 

2  P  {y' COS  y  -z' COS  13)  =  2  m  (/ ~  -  s' •  ^^  ; 


(255) 


188         ELEMENTS    OF    ANALYTICAL    MECHANICS. 
given  by  the  principle  of  the  centre  of  inertia,  we  find 

dx 
2  P  cos  a  =  — ^  .'Lm: 
d  t 


2  P  cos  /3  =  ^  •  2  m ; 


(251) 


dz, 


2  P  cos  7  —  -jj'lm; 
and  substituting  M  for  2  m,  we   have 


ffrf 


,  ^  '^  c^  ~ 


■which  are  wh 
of  the  body, 
centre  of  ine: 
trated  in    it,  and  the   forces   applied  immediately  to  that  point. 

g  186. — Replacing  the  first  members  of  the  above  equations  by 
their  values  given  in  Equations  (41),  and  denoting  by  V  the  velocity 
which  the  resultant  R,  can  impress  upon  the  whole  mass,  then  will 

2Pcosa  =  if  Fcosa;    2P  cosi  =  i/Fcos  5  ;    2  P  cose  =  i/"  If  cose; 

substituting   these   above,  we  find 

d  X, 


V.  cos  a=: 
V . cos  b  = 
V .  cos  c  = 


dt  ' 
dt  ' 
dt    ' 


(252) 


MECHANICS    OF    SOLIDS. 


189 


and  integrating, 

X,  =  V'cosa.t  +  C\    "^ 

y^  =  V.  cos  b.i  +  C",    I (-253) 

z,  =  V.  cos  c.i  +  C",  _ 

and   eliminating   t   from   these   equations,    V  will   also  disappear,   and 
we  find, 


2/  =  3//  • 

y/  =  ^/  • 


cos  c 
COS  a 

cos  c 
cos  6 

COS  b 


C  cos  c  -  C"  cos  a     ^ 


cos 

a 

C" 

COS 

c  — 

C" 

cos 

b 

cos 

b 

C" 

cos 

b  - 

C" 

cos 

a 

cos  a 


(254) 


which  being  of  the  first  degree  and  either  one  but  the  consequence 
of  the  other  two,  are  the  equations  of  a  straight  line.  This  line 
makes  with  the  axes  x,  y,  z,  the  angles  a,  5,  c,  respectively,  and  is, 
therefore,  parallel   to   the   resultant  of  the  impressed  forces. 

Whence  we  conclude,  that  the  centre  of  inertia  of  a  body  acted 
upon  simultaneously  by  any  number  of  impulse  forces,  will  move 
uniformly  in  a  straight  line  parallel  to   their   common  resultant. 


MOTION   ABOUT   THE   CENTRE   OF   INERTIA. 

§  187. — Substituting,  in  Equations  (249),  for  dx^  dy  and  dz^  their 
values   from    Equations    (34),    reducing   by 

2  m  x'  =  0, 

2  m  y'  =  0, 

2  m  z'  =  0, 
and  we  find, 

2P  (x'cos/3  -  y'cos  a)  =  2  ni  (x'-  -'^  -  /  — ) 

^         d  t  d  t  ^ 


2  F  (^z'  cos  a  —  x'  cosy)  =  2  ?/i  (z''  — '- x'  ■—)  ; 


-atr.^j.". 


-R^.-^^-z 


dx' 
dt 

dz' 


2  P  (y''cos  7  -  z'  cos  /5)  :=  2  m  L' .  I'-l  _  z' .  ~f)  ; 

^         d  t  d  t  ^ 


(255) 


190 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


whence,  the  motion  of  the  body  about  its  centre  of  inertia  will  be 
the  same  whether  that  point  be  at  rest  or  in  motion,  its  co-ordinates 
having   disappeared   entirely  from  the  equations. 


AJSTGULAK   VELOCITY. 

§188, — Substituting,  in  Equations  (255),  for  dx\  dy'  and  dz\  their 
values  -afe-  gi?^  i>y  Eqojatibits-  (Sll^  re^«^ing-  by  the  relations, 

c/i^'  =^    d-]..z'  =  —  d(^  .y\ 

el  t^''  -      c?  9  .  .t;'  =  —  d-ui  .  z\ 

d:^'-      dzi  .7/  =  —  d-^.x\ 

obtained  from  Equations  (35),  (3G),  (37),  and  replacing  the  first  mem- 
bers of  Equations  (255)  by  £^,31^  and  i\^,  respectively,  §175,  we  have 

dcp 


-^  .  2  m  (a;'2  +  y"^)  =  L, 


d\ 
~dt 

dzi 
dt 


whence 


•  2  m  (y'2   -}_  2'2)    _  ^ 

J  (p  L, 

ITt    ~  Em.  (x'2  +  y'2)' 

(14.  _  M, 

dt    ~  2  m  .  (j;'2  -f  z'^)  ' 

d  vi  JV, 


(256) 


(257) 


di  2m.  (y'2  -f-  z'^) 

That  is,  the  component  angular  velocity  about  either  axis,  is  equal  to 
the  moment  of  the  impressed  forces  divided  by  the  moment  of  inertia 
with  reference  to  that  axis. 

d  s 

The    resultant   angular    velocity   being    denoted    by    — —-'  we   also 


have 


ds,  _    l_     / 
dt         dtV  ' 


d(p'^  -f  d-].^  +  dzs^ 


(258) 


MECHANICS    OF    SOLIDS. 


191 


§  189. — Tlie   axis   of  instantaneous   rotation   is   found  as   in   |  171, 
by  making  in  Equation  (219), 

dx'  =  0;  dy'  =  0;  dz'  z=  0; 
Avhich  gives,  Equations  (220)  and  (218), 


.'    -    y>  .^'     .' 


d-m 


d  -Uj' 


(259) 


which  are  the  equations  of  a  riglit  line  through  the  centre  of  inertia. 


AXIS   OF   SPONTANEOUS   KOTATION. 
§  190. — If  both  members  of  Equations  (34)  be  divided  by  d  t,  we 
have 

d  X         d X,         dx' 
d  t  d  t  d  t 

^  _  ^y^  ,    dx'  . 

dt  ~    dt  dt   ^ 

d  z        dz,  d  z' , 

dt  d  t  dt 

and  if  for  any  series  of  elements  we  have 


d  X  d  y 

dJ  ~      '   TT 


^       dz 
0;    -^-  =  0: 
'     d  t 


then  will 


d  X, 
~dT 


d  x' ,    d  y,  d  y'  ,    d  Zi 

d  t        d  t  d  t        d  t 


dz' 
Tt 


(2G0) 


(261) 


dx'      dy' 
'dJ''      dt 


and    substituting:     for   ^V^,     -^,    and    -^^,     their    values    given    in 


dz' 
ITt 


Equations  (217),  and  for  -— ->    -y^'j    and  -—-?   their  values   given   by 
Equations  (252),  we  find 


z'  =y 


y   —X 


,    d  !^  V .  cos  a , 

d-^  d  4^ 

d  t 
,    d(p  F.  cos6. 

dzi  d  zs 

dt 
,    d-l^         F.eosc 


d-us 


dm 
dt 


(262) 


192  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Which  are  the  equations  of  a  right  line  parallel,  Equations  (259),  to 
the  instantaneous  axis. 

This  line  is  called  the  axis  of  spontaneous  rotation  ;  because,  being 
at  rest,  Equations  (260),  while  the  centre  of  inertia  is  in  motion,  the 
whole  body  may  be  regarded,  during  impact,  as  rotating  about  this 
line.  Its  position  results  from  the  conditions  of  Equations  (261), 
which  are,  that  the  velocity  of  each  of  its  points,  and  that  of  the 
centre  of  inertia  must  be  equal  and  in  contrary  directions.  Tlie  dis- 
tinction between  the  axes  of  instantaneous  and  of  spontaneous  rota- 
tion is,  that  the  former  is  in  motion  with  the  centre  of  inertia,  while 
the  latter  is  at  rest. 

The  relative  positions  of  the  axis  of  spontaneous  rotation,  and 
the  line  along  which  the  resultant  impact  acts,  cannot  be  affected 
by  any  change  in  the  co-ordinate  axes.  For  simplicity,  take  the 
fixed  axes  so  that  the  movable  axis,  x'  shall  be  parallel  to  the 
line  of  the  resultant,  and  the  plane  x'  y'  shall  contain  this  latter  line. 
In  this  case, 

cos  a  =  1  ;     cos  5  =  0;     cos  c  =  0  ; 

and   the  values  of  y'  and  z\  in  L,  M,  N^  will  be, 

y'  =  ^/ ;    2'  =  0 ; 

in  which  e,  is  the  perpendicular  distance  from  the  centre  of  inertia 
to   the  direction  of  the   resultant  impulse. 
Also,  in  Equations  (254),  we   shall  have, 

cos  c        „       cos  h 

=  0  ;     =  0  ; 

cos  a  cos  a 


in   Equations  (257), 


c?(p  —Re, 


d  t  2  m  .    (.C'2    -f    y'2)    ' 

ii^O-     —  -0- 
dt  '      dt    ~     ' 


and,  Equations  (262), 


d(n  dcD 

d-\,  ^      dzi 


MECHANICS     OF    SOLIDS.  193 

whence,    the    axis   of    spontaneous   rotation    is    perpendicular   to    the 
direction  of  the  I'esultant  impulse,  a '•  i  ts  c-.'jse.'nrntiv  ,/,/>  cix?i  u-  x.. 

From  the  first  and  second  of  Equations  (2G2),  we  have  by  clear- 
ing  the  fractions, 

'•-dT  =  y'-dT-  ^' 

2'.  —  =  a:'.  -^• 
dt  dt    ' 

and  since   (^4/  =  0;    dvs  =  0;  we  obtain,  after  substituting  the  value 

of  ^,  that  of  Ji  =  M  V,  and   that  of  2  w .  (.c'2  +  y'2)  =  Mk;^, 

x'  =  0;     y'  =A' (263) 

The  axis  of  spontaneous  rotation  is,  therefore,  in  the  plane  z'  y', 
and  cuts  the  perpendicular  drawn  from  the  centre  of  inertia  to  the 
line  of  the  resultant,  and  at  a  distance  from  that  centre  equal  to 
the  square  of  the  principal  radius  of  gyration  with  reference  to  the 
axis  of  instantaneous  rotation,  divided  by  the  distance  of  the  line 
of  the   resultant   from    the    same    centre. 

Adding  e^  to  both  members  of  the  second  of  the  above  equations, 
we   have,  after  writing  e  for  y\ 

e  +  e^  =  i^^^  =  l (2G4) 

in  which  I  is   the   distance  from  the  axis  of  iaiSiSSftlancous   rotation  to 

A 

the   line  of  the   resultant  impact. 

§  191. — The  body  being  perfectly  frep,  and  the  axis  of  spontaneous 
rotation  at  rest  while  the  other  parts  of  the  body  are  acquiring 
motion,  it  is  plain  that  the  forces,  both  impressed  and  of  inertia,  are 
so  balanced  about  that  line  as  to  impress  no  action  upon  it.  The 
points  of  the  body  on  the  line  of  the  resultant  impulse  are  called 
centres  of  percussion  in  reference  to  the  axis  of  spontaneous  rotation. 
A  centre  of  percussion  is,  therefore,  any  point  at  which  a  body  mav 
be  sti'ucl^  in  a  direction  perpendicular  to  the  plane  through  tbe  centre 


194  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

of  inertia  and    axis    of   spontaneous    rotation,  M'ithout   communicating 
any   shock   to  a  physical   line   coincident  in  position   with   that  axis. 

g  192. — In  Equation  (258),  we  have 

els  dp  Re,  _    Ve^ 


dt  dt         2  m  (a;'2  +  y'2)  k^ 


(265) 


§  193, — If  r  denote  the  distance  of  the    elementary  mass  wi,  from 
the  axis  of  z',  the  velocity  of  rotation  of  this  element  will  be 

d  (p 

and  its  centrifugal  force 

d  (p^     r^  VI        d  (p2 

de       r  dt''  ' 

which  is  the  pressure    exerted    by  the  inertia  of  m  upon    the  axis  z\ 

this  axis  being  that  of  instantaneous  rotation.     Its  point  of  application 

is  on  that  axis.     The  cosines  of  the  angles  which  its  direction  makes 

x'  y' 

with  the  axes  x'  and  y'  are  respectively  —  and  —  ;    its    components 

in  the  direction  of  these  axes,  are,  therefore, 

C?(p2  d(p"        , 

—V  '  x'  -m    and    ——-  •  y'  -m, 
dfi        ■  dfi    ^        ^ 

and  its  moments  in  reference  to  the  axes  x'  and  y',  are 

'   df"  ,    ,         .     dcp^ 

-—— -  -m-  X  z     and    -^^ ' 
d  t-  d  t^ 

and   the   sum    of   the   moments   of  the   centrifugal  forces   of  all   the 
elements  of  the  body  in  reference  to  these  axes  are 

d'lP'  ,   ,         ^    d(p^     ^  ,   , 

--^  .  i;  m  •  x'  z'    and    -^  -Hm-y'  s' . 
df  dfi^ 

Now  if  the  instantaneous  axis  be  also  a  principal  axis,  then  will 

2  m  .x'z'  =  0,    and    Imy'  z'  =  0  ; 
and  there  will   be   no   pressure   on  the  instantaneous  axes.      If,  there- 


MECHANICS     OF    SOLIDS. 


195 


fore,  the  impressed  force  be  so  applied  as  to  cause  the  body  to 
begin  to  rotate  about  a  principal  axis,  the  rotation  will  continue 
about  this  axis,  and  the  axis  is  said  to  be  permanent;  but  if  the 
rotation  do  not  begin  about  a  principal  axis,  the  axis  of  rotation 
will  change  its  position  under  the  pressure  arising  from  the  centrifu- 
gal forces  developed,  and  this  change  -will  continue  till  the  position 
of  the  axis  of  rotation    reaches  that    of  a   principal    axis. 

MOTION   OF   A   SYSTEM   OF   BODIES. 


§194. — "VVe  have  seen  that  the  Equations  (117)  and  (119)  give 
all  the  circumstances  of  motion  of  the  centre  of  inertia  of  a  single 
body  in  reference  to  any  assumed  point  taken  as  an  origin  of  co- 
ordinates. For  a  second,  third,  and  indeed  any  number  of  bodies, 
referred  to  the  same  origin,  we  would  have  similar  equations,  the 
only  difference  being  in  the  values  of  the  co-ordinates,  of  the  inten- 
sities and  directions  of  the  forces,  and  of  the  magnitudes  of  the  masses. 
This  difference  being  indicated  in  the  usual  way  by  accents,  we  should 
obtain  by  addition, 


2  M- 


^.M- 


2.i/. 


(P  X 

d  C- 

(Py_ 
dt~ 


=  2  A"; 


=  2  1' 


=   2  Z 


(266) 


df^ 


(267) 


in  which  it  must  be  recollected  that  x,  y,  2,  drc,  denote  the  co- 
ordinates of  the  centres  of  inertia  of  the  several  masses  i/,  &c., 
referred   to   a  fixed  origin. 


196 


ELEMENTS     OF     AIS'ALYTICAL    MECHANICS. 


MOTION    OF    THE    CENTRE    OF   INERTIA    OF   THE    SYSTEM. 

§195. — Taking  a  movable  origin  at  the  centre  of  inertia  of  the 
entire  system,  denoting  the  co  ordinates  of  this  point  referred  to 
the  fixed  origin  by  x^,  3/^ ,  2^ ,  and  the  co-ordinates  of  the  centres 
of  inertia  of  the  several  masses  referred  to  the  movable  origin  by 
x',  y\  z',  &c.,  we  have,  the  axes  of  the  same  name  hi  the  two  sys- 
tems being  parallel, 

X  =  x^  -\-  x\ 

y  =  yy  +  y', 

and, 

rf2  X  =  d^x^  +  cP  x',  ^ 

d^tj  =  d^^2j,  +  d'y',   [ (268) 

d^-z  ^d^z^  +  d^-  z',  j 

which  substituted   in  Equations  (266),  and  reducing  by  the  relations, 
.    ^*-'=*„    iM.d^x'  =  0;     ^3fd^y'  =  0;     ■EMd^z'  =  0;    •    -(260) 
obtained  from    the   property  of  the    centre  of  inertia,  wc  find 
d^x, 


dC- 

dfi 

d^z 
If 


(270) 


which  being  wholly  independent  of  the  relative  positions  of  the  several 
bodies,  show  that  the  motion  of  the  centre  of  inertia  of  the  system 
will  be  the  same  as  though  its  entire  mass  were  concentrated  in 
that  point,  and   the  forces  applied  directly  to  it. 


§196. — Multiplying  the  first  of  Equations,  (^Si^V)  ^7  Vn  the  second 


MECHANICS     OF    SOLIDS. 


197 


by  x^,  and  taking  the  diflVrcnce ;  also,  their  first  by  z^  the  third 
by  a-^,  and  taking  the  difference,  and  again  the  second  by  ^^,  the 
third   by  y„    and    taking    the   difference,    we  find 


rf2 


>-       (271) 


which  will  make  known  t-hc    circumstances  of  motion  of  the  common 
centre   of  inertia   about   the  fixed   origin. 

INEKTIA. 

as   given  by 
ations  (271) 


(272) 


liquations  Irom  winch  ail  traces  ot  tiic  position  ot  the  centre  of 
inertia  have  disappeared,  and  from  which  we  conclude  that  the 
motion  of  the  elements  of  the  system  about  that  point  will  be  the 
same,  whether  it  be  at  rest  or  in  motion.  These  equations  are 
identical  in  form  with  Equations  (118);  whence  we  conclude  that 
the  molecular  forces  disappear  from  the  latter,  and  cannot,  there- 
fore, have  any  influence  upon  the  motion  due  to  the  action  of  the 
extraneous  forces. 

CONSERVATION   OF   THE   MOTION   OF  THE  CENTRE    OF   INERTIA. 

g  198. — If  the  system  be  subjected  only  to  the  forces  arising  from 
the   mutual   attractions   or   repulsions  of  its   several   parts,   then   will 

1  X  =  0;  ZY=0;  2Z  =  0. 


196    -       ELEMENTS     OF    AIS-ALYTICAL    MECHANICS. 


MOTION    OF    THE    CENTRE   OF   INEETIA    OF   THE    SYSTEM. 

8  195, — Taking  a  movable  origin  at  the  centre  of  inertia  of  tiie 
entire  system,  denoting  the  co  ordinates  of  this  point  referred  to 
the  fixed  origin  by  x^ ,  y^ ,  z^,  and  the  co-ordinates  of  the  centres 
of  inertia  of  the  several  masses  referred  to  the  movable  origin  by 
a;',  ?/',  z',  &c.,  we  have,  the  axes  of  the  same  name  in  the  two  sys- 
tems being  parallel, 

X   ^n   X     -\-   X  , 

y  =  y^  +  y\ 
I   „/ 

a^'3'  yur^u^    ^       z:^       ^^  •  ^  J^^^,  )   -^^  ^-f^^^^^-^ 

which  substi  >^:^  '^^  "-  -^^-^^         ^^    -^  ^^-^ 

^^^^'.o.TrJf.,.:     -^^.^^'  '^^^  ^'i^  ^^^.-^   Yr'  -r  X^,-r, 

obtained  froi  s:^K;?cy^^  ^^^S^^^^,'^^jxi^  -  o  „  z  c^^^' ^^IJLs.  =^.,-<*-" 


d?z, 
'd¥ 


(270) 


which  being  wholly  independent  of  the  relative  positions  of  the  several 

bodies,  show  that  the  motion  of  the    centre  of  inertia  of  the   system 

will   be  the   same   as   though   its    entire   mass   were   concentrated   in 

that  point,  and   the  forces  applied  directly  to  it. 

fi  70) 
§190. — Multiplying  the  first  of  Equations,  ("Si^^"^,  by  y ,,  the  second 


MECHANICS     OF    SOLIDS, 


197 


by  a:,,  and  taking  the  difference ;  also,  their  first  by  z^  the  third 
by  X,,  and  taking  the  difference,  and  again  the  second  by  2-^,  the 
third   by  y„    and   taking   the   difference,   we  find 

which  will  make  known  Uic  circumstances  of  motion  of  the  common 
centre    of  inertia   about   the  fixed   origin. 

MOTION   OF   THE   SYSTEM   ABOUT   ITS   COMMON   CENTRE   OF   INERTIA. 

§197. — Substituting  the  values  of  d- x^  d^  ij,  and  d^  z,  as  given  by 
Equations  (268),  in  Equations  (2G7)  and  reducing  by  Equations  (271) 
and   (269),   there   will   result 

d'^  y'  .    d'^  x''^ 


/        d^  y  ,    d'^  X  \  ,  ^T    , 

\        dt^  dt-y  ^ 


Xy') 
Zx') 


^^^■{y'-%-^'-t0=^('^'-^-^\ 


(272) 


Equations  from  which  all  traces  of  the  position  of  the  centre  of 
inertia  have  disappeared,  and  from  which  we  conclude  that  the 
motion  of  the  elements  of  the  system  about  that  point  will  be  the 
same,  whether  it  be  at  rest  or  in  motion.  These  equations  are 
identical  in  form  with  Equations  (118);  whence  we  conclude  that 
the  molecular  forces  disappear  from  the  latter,  and  cannot,  there- 
fore, have  any  influence  upon  the  motion  due  to  the  action  of  the 
extraneous  forces. 

CONSERVATION    OF   THE   MOTION    OF   THE  CENTRE    OF   DTERTIA. 

§  198. If  the  system  be  subjected  only  to  the  forces  arising  from 

the   mutual    attractions   or   repulsions  of  its    several    parts,    then   will 

2  A"  =  0 ;  2  r  =  0  ;  2  Z  =  0. 


198  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

For,  the  action  of  the  mass  M^  upon  a  single  element  of  iT, 
will  vary  with  the  number  of  acting  elements  contained  in  M\ 
and  the  effort  necessary  to  prevent  M'  from  moving  under  this 
action  will  be  equal  to  the  whole  action  of  M  upon  a  single  element 
of  M'  repeated  as  many  times  as  there  are  elements  in  M'  acted 
upon ;  whence,  the  action  of  M  upon  M'  will  vary  as  the  product 
MM.  In  the  same  way  it  will  appear  that  the  force  required  to 
prevent  M  from  moving  under  the  action  of  M\  will  be  propor- 
tional to  the  same  product,  and  as  these  reciprocal  actions  are 
exerted  at  the  same  distance,  they  must  be  equal;  and,  acting  in 
contrary  directions,  the  cosines  of  the  angles  their  directions  make 
with  the  co-ordinate  axes,  will  be  equal,  with  contrary  signs.  Whence, 
for  every  set  of  components  P  cos  a,  P  cos  ^,  P  cos  7,  in  the 
values  of  2  X,  2  y,  2  Z,  there  will  be  the  numerically  equal  com- 
ponents, —  P'  cos  a',  —  P'  cos  /3',  —  P'  cos  7',  and.  Equations  (270), 
reduce,    after    dividing   by    2  if,    to 

S  =  «-'  '^  =  ''>  ?*  =  »•  •  •  ■  (^^^) 

and   from  which  we   obtain,  after   two  integrations, 

X,  =  C'.t  +  D';    ^ 

y,  =  C"  t+  D";    I (274) 

z,  ==  C"  t  +  D'" ;  J 

in  which  C",  C",  C",  D\  D"  and  D'"  are  the  constants  of  inte- 
gration; and  from  which,  by  eliminating  t,  we  find  two  equations  of 
the  first  degree  between  the  variables  x^ ,  y,,z^^  whence  the  path 
of  the    centre    of  inertia,  if  it   have   any  at   all,  is    a   right   line. 

Also   multiplying   Equations  (273)  by  %dx,,  "^dy^,  2dz^,  respec- 
tively, adding  and   integrating,  we   have 

dx^^  ^dy.'^  +  dz,^  ^  72  ^  (7     .     .     .     .     (275) 
dfi 

in  which  C  is  the  constant  of  integration  and  V  the  velocity  of  the 
centre  of  inertia  of  the  system.  From  all  of  which  we  conclude, 
that   when  a  system   of    bodies   is    subjected    only  to    forces   arising 


MECHANICS     OF    SOLIDS. 


199 


from  the  action  of  its  elements  upon  each  other,  its  centre  of  inertia 
will  either  be  at  rest  or  move  uniformly  in  a  right  line.  This  is 
called   the   conservation  of  the  motion  of  the   centre  of  inertia. 


CONSEKVATIOX    OF    AREAS. 

g  199.— The   second  member   of  the  first  of  Equations  (-272)  may 
be  written, 

Yx'  -  Xy'  +Y'  x"  -  X'y"  +  &c. ; 

and   considering   the   bodies   by  pairs,  we  have 

X  =  -  A" ;     r  =  -  F' ; 

and  eliminating  X'  and   Y'  above  by  these  values,  we   have 

Y{z'  -x')  -X{y'  -  y")  +  &c. 
But, 


V        '  P 

in  which  p  denotes  the    distance   between    the   centres    of    inertia   of 
the  two  bodies.     And  substituting  these  above,  we  get 

P .  ?^-:=-^  {x'  -  X")  -  P  • (/  -  /')  =  0  ; 

2)  P 

and  the   same   being   true   of  every   other  pair,  the  second   members 

of  Equations  (272),  will   be  zero,  and   we   have 


and  integrating 


_^   x'  d  y'  —  y'  d  x'        ^, 


2if. 


dt 

z'd  x' 

-  x' 

d  z' 

dt 

y'  d  z 

-  z 

dy' 

dt 


=  C". 


(276) 


200  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

But  §  156,  x'  dy  —  y'dx,  is  twice  the  differential  of  the  area  swept 
over  by  the  projection  of  the  radius  vector  of  the  body  M,  on  the 
co-ordinate  plane  x\  y',  and  the  same  of  the  similar  expressions  in 
the  other  equations,  in  reference  to  the  other  co-ordinate  planes ; 
whence,  denoting  by  A^,  A^,  A^.,  the  whole  areas  described  in  any 
interval  of  time,  t,  by  the  projections  of  the  radius  vector  of  the  body 
M,  on  the  co-ordinate  planes,  x'  y',  x'  z\  and  y'  z' ;  and  adopting 
similar  notations  for  the  other  bodies,  we  have 

d  A^ 
dt  ' 

dt  ' 

2  M-  i^  =  C"; 
dt  ' 

in  which  C",  C",  C",  are  twice  the  sums  of  the  areas  swept  over  in 
a  unit  of  time  by  the  projections  of  the  radii  vectores  on  the  planes 
x'  y\  x'  z',  and  y'  z' ;  and  by  integration  between  the  times  t^  and  f, 
giving  an  interval  equal  to  t, 

:s.M.A^^  C'.t;    ' 
2M.A^=   C"  t; 
^M.A^=   C"  t; 

whence  we  find  that  when  a  system  is  in  motion  and  is  only  sub- 
jected to  the  attractions  or  repulsions  of  its  several  elements  upon 
each  other,  the  sum  of  the  products  arising  from  multiplying  the 
mass  of  each  element  by  the  projection,  on  any  plane,  of  the  area 
swept  over  by  the  radius  vector  of  this  element,  measured  from 
the  centre  of  inertia  of  the  entire  system,  varies  as  the  time  of  the 
motion.     Tliis  is  called   the  principle    of  the   conservation  of  areas. 

§  200. — It  is  important  to  remark  that  the  same  conclusions 
would  be  true  if  the  bodies  had  been  subjected  to  forces  directed 
towards  a  fixed  point.  For,  this  point  being  assumed  as  the  origin 
of  co-ordinates,  the  equation  of  the  direction  of  any  one  force,  say 
that   acting   upon  M,  will   be 

Tx  —  Xy  =  0; 


MECHANICS    OF    SOLIDS.  201 

and  the  second  members  of  Equations  (207)  will  reduce  to  zero ; 
and  the  form  of  these  equations  being  the  same  as  Equations  (272), 
they  will   give,    by  integration,  the    same    consequences. 

INVARIABLE    PLAN'E. 

dy' 
§201. — If  we  examine  Equations  (27G),  we  shall  faid  that  ^f  ■  —r- 

is  the  quantity  of  motion  of  the  mass  J/,  in  the  direction  of  the 
axis   y\  and    is  the   measure  of  the  component  of  the   moving  force 

d  x'     . 
in   that  direction  ;    the   same    may  be  said   of  M  •  -y— >    in  the  direc- 
tion   of  the    axis   x' ;  whence    the    expression, 
,,   dy' x'  —  d x'  y' 

is  the  moment  of  the  moving  force  of  M^  with  respect  to  the 
axis  z' .  Designating,  as  before,  the  sum  of  the  moments  with  respect 
to  the  axes  z\  y'  and  x',  by  L, ,  31, ,  iV^ ,  respectively,  Equations  (276) 
become 

L,  =  C;     M,  =  C";     N,  -C". 

Denoting   by  A^  B   and    C,    the   angles   which  the   resultant   axis 
makes   with   the   axes  z\  y'  and  x\  we   have,  §110, 


cos 

JX    — 

vw 

+  M,-^ 

+  iV^2 

B  = 

M, 

VA^ 

^M^ 

+  iV^/ 

COS 

C  = 

N, 

(J^ 

C" 


(277) 


These  determine  the  position  of  the  resultant  or  principal  axis. 
The  plane  at  right  angles  to  this  axis  is  called  the  principal  plane. 
The  position  of  this  plane  is  invariable,  and  it  is  therefore  called 
the  invariable  plane,  cither  when  the  only  forces  of  the  system  are 
those  arising  from  the  mutual  actions  and  reactions  of  the  bodies 
upon  each  other,  or  when  the  forces  are  all  directed  towards  a  lixed 
centre. 


202  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

PEINCIPLE   OF    LIVESTG    FOKCE. 

8  202. The   components  of  the  extraneous  force,  in   the   direction 

of  the  axes,  impressed   upon   the  element  7n  of  a  mass  M,  are 

P  cos  a  =  X;     P  cos  ^  =:  F;     P  cos  7  =  .^ ; 

the   components   of    the  inertia  of  m   in   the  directions  of  the  same 


axes  are 


"'■Tf'^    "^"dT-'^    "^'U 


and   the  resultant   components   in   these    directions  are 
d'^x        ^^  d^y        ^  d'^z 

^-"^'dTf'^    ^-"^'rfl^'    ^-^-7^' 

and  their  virtual  moments, 

and  similar  expressions  for  the  other  elements  ?n',  vi",  &iG.  But 
these   must  be  in    equilibrio ;    whence, 

.|(.r-..-),.+  (r-./;|)..+  (.-..S)*4=«^ 

in  which  S  x,  S  y  and  5  z,  are  small  spaces  described  by  m,  in  the 
direction  of  the  axes,  consistently  with  the  connection  of  the  parts 
of  the  system  one  with  another   at   the    time  i. 

But  the  spaces  actually  described  at  the  end  of  the  time  t,  being 
consistent  with  the  connection  of  the  parts  of  the  system  one  with 
another,  we   have, 

dx  ^  dy  dz      . 

dt  '       -^  dt  '  dt  ' 

which   in  the   above   equations  give,  after  transposition, 

j  dx  d?x_       dy^    d^       dz^   d^ )  { y  ^ty  ^   ,7  ^\ 

^^^'ldt'dfi'^dt'dP^di'dfi\~      V'dt'^     'dt'^    'dtr 


MECHANICS     OF    SOLIDS.  203 

which  becomes,  by  integrating, 

\dfi  ^ dfi  ^ df'  \       J  \    di^     dr     dt/    ^  ' 

or   replacing  the  first   member   by   its  value, 

^'-^  =  2^/  (a-  %  +  r^{  +z^a,  +  c...  (2-8) 

g203, — If  P  be  the  mutual  pressure  of  two  elements  7?i  and  m\ 
in  contact^  at  the  point  whose  co-ordinates  are  x,  y,  z,  then  the  ex- 
pression 

/(-^•4:+^'-'^^-^')--  •  •  •  (-») 

for   the   element   m  becomes 

/^  /  dx  ^    dy    ^  dz\ 

p(cosa.-  +  cos/3.-  +  cosr.-;.7^  ; 

and  for  m',  it  becomes 

/•^/  d X     ^  ^    dy    ,  dz\ 

-  yp  (cos  a.— +  cos /3.  _  +  cos  7. -;  cf^ 

and  their    sum  will   be  zero;  therefore  the  pressure  F  will  disappear 
from  Equation  (2T8). 

If  the  elements  ??i  and  m'  be  not  in  contact,  but  be  separated 
by  the  distance  r,  let  xy  z  and  x'  y'  z'  be  their  respective  co-ordinates, 
P  their  mutual   action,  supposed  some  function  of  r ;    then  will 

/p  x'  y  —  11'  Z  —  z' 

cos  a  =  ;     cos  /3  =  '—  ;     cos  y  =  j 


X  —  X              ^,            y  —  y                ,             z  —  z 
cos  a'  =  —  ■ ;    cos  /3'  =: ;    cos  7'  = 


and   the    expression  (279)  for  the  element  ?»,  will    be 


204         ELEMENTS     OF     ANALYTICAL    MECHANICS, 
and  for  the   element  in\ 

r>      (x-x'     dx'     ,     y-y'     dy'      ^    z-z'      dz'\ 

-J^\rv~"dJ  +  "^'^  +  ^^■"77/'^'' 

and  P  -will  appear  in  Equation  (278)  under    the  form, 

and  since 

.r2  =  (x  -  x')2  +  (y  -  y'f  +  (z  -  z'f, 
by  differentiating 

rdr  =  (x  -  x')^  (,r  -  a;')  +  (y  -  y>?  (y  -  y')  +  (2  -  z')d{z  -  z')', 
and  the  above  reduces  to  2fFdr,  which  in  Equation  (278),  gives 

2mi;2  ^  2  2/PJr  +   C. (280) 

Any  force  which  acts  upon  a  fixed  point,  will  not  appear  in  tlie 
equation  of  the  living  force,  since  the  velocity  of  such  point  is  zero. 
Neither  will  the  force  of  rolling  friction,  where  one  of  the  bodies 
is   fixed.     The  force    of  sliding  friction   will. 

If  the  forces  act  upon  none  of  the  elements  of  the  system  except 
such  as  remain  invariably  connected  during  the  motion,  the  living 
force  must  remain  the  same  throughout  the  motion,  for  in  this  case 
dr  =  0,  will  give.  Equation  (280), 

2  jn  v"^  =  C. 

This  is  called   the  principle  of  the   conservation   of  living  force. 

The  value  of  P,  being  by  hypothesis  a  function  of  r,  Avill 
always  be  integrable.  The  integration  being  performed,  and  r  being 
replaced  by  its  value  in  functions  of  x  y  z,  x'  y'  z',  and  the  same  for 
r'  &c.,  the  living  force  of  the  system  will  become  a  function  of  the 
co-ordinates  of  the  bodies'  places,  and  when  taken  between  limits 
will  be  dependent  alone  upon  the  co-ordinates  of  the  first  and  last 
positions  of  the  bodies,  and  wholly  independent  of  the  paths  described 
by    them. 


MECHANICS     OF     SOLIDS,  205 

1 204. — The  co-ordinates  of  the  clcnicnt  ???,  at  the  end  of  the 
time   t,  being   x  7/ z,  we  have 

V-  =  —  {d  X"  +  d  ij-  -}-  dz-), 

and  substituting  for  dx,  dy^  dz,  their  values  obtained  from  Equa- 
tions (2GS), 

z;2  =  -L,  [dx,^  +  dy;^  -f  dz^')  +  ^  {dx^dx\+  dy,.dy'^  dz^  dz') 
+  ^{dx'-^  +  dy''^+dz'% 

substituting  this  in    Equation  (2T8)  and  reducing  by  the  relations, 

^mdx'  —  0;    ^mdy'  =  0;    ^  m  d  z'  —  0, 
obtained   from    the    property    of  the    centre    of  inertia,  we   find 

2m  y2  izz   F,2  2  7?i  +  2  m  i;'2  ; (281) 

in  which  V^  denotes  the  velocity  of  the  centre  of  inertia  of  the  entire 
system,  and  v'  that  of  each  element  about  that  centre.  Whence,  th(^ 
living  force  of  a  material  system  in  motion  is  equal  to  the  living 
force  arising  from  the  motion  of  translation  of  the  centre  of  inertia, 
increased  by  the  living  force  arising  from  the  motion  about  the 
common    centre   of  inertia   of  the   whole. 

205. — Differentiating  Equation  (280),  we  have 

oj:Pdr  =  ^-^^.dt; (282) 

and  if,  at  any  instant  during  the  motion  the  living  force  become 
a  maximum  or  minimum,  then  will  1.Pdr  =  0,  and  the  systena  will. 
Equation  (28),   be  in  equilibrio. 

Also,  §  134,  when  the  living  force  is  a  maximu7n.,  the  position 
which  the  system  assumes  would  be  that  of  stable  equilibrium,  if  all 
the  velocity  were  destroyed;  and  when  the  living  iovw  is  a  minimum^ 
the  position  would  be  one  of  unstable  equilibrium.  And  since  a  func- 
tion passes  through  its  maximum  and  minimum  Values  alternately,  as 
the    variable    increases    continuously,    the     system    when    in    motion 


206  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

will  pass  through  the  positions  of  stable  and  tmstable  equilibrium 
alternately. 

§  206. — If,  during  the  motion,  two  or  more  bodies  of  the  system 
impinge  against  each  other  so  as  to  produce  a  sudden  change  in  their 
velocities,  the  sum  of  the  living  forces  will  undergo  a  change.  To  esti- 
mate this  change,  let  A,  B,  C  be  the  velocities  of  the  mass  wi,  in  the 
direction  of  the  axes  before  the  impact,  and  a,  5,  c  what  these  veloci- 
ties become  at  the  instant  of  nearest  approach  of  the  centres  of 
inertia  of  the  impinging  masses,  then  will 

A  —  a^    B  —  b,     C  ^  c, 

be  the  components  of  the  velocities  lost  or  gained  by  ??i  at  the  instant 
corresponding   to   this   state  of  the   impact,  and 

m{A  —  o),    m{B  —  b),    971(0  —  c), 

the  components  of  the  forces  lost  or  gained.  The  same  expressions, 
with  accents,  will  represent  the  components  of  the  forces  lost  or 
gained  by  the  other  impinging  bodies  of  the  system.  These,  by 
the   principle   of  D'Alembert,  §  71,  are  in  equilibrio,  whence 

I,m{A  —  a)  5 X  -{-  i:7n{B  —  b)  0  7/  -{-  I,7n{C  —  c)Sz  =  0. 

The  indefinitely  small  displacements  S  x,  S  y,  S  z,  &c.,  must  be  made 
consistently  with  the  connection  by  virtue  of  which  the  velocities  are 
lost  or  gained ;  but  as  a,  b,  c  denote  the  components  of  the  actual 
velocities  of  any  two  bodies  during  the  time  of  greatest  compression, 
when  alone  these  velocities  are  equal,  this  condition  will  be  fulfilled 
if  we  make 

8  X  =i  a  .0  t\      S  y  z:^  b  .S  t;      S  z  =:  c .  S  t. 

These  values  being  substituted  in  the  above  equation,  we  have, 
after   dividing   by  d  t, 

^m{A  —a)a-{-:Em(B  —  b)b-^i:m{C—c)c  =  0  •  •  (283) 


or. 


Sm{Aa  -\-  Bb  -{-  (7c)  -  2m  (a^  +  b^  -\-  c^)  =  0   •    •  (284) 


MECHANICS     OF    SOLIDS.  207 

But   we   have   the   identical   equation, 

(A-ar-  +  (B-,f+(C-cY  =  \       ^^  +  ^'+C-'  +  a'  +  l,' 

^  ^         ^  ^         ^  '         (  +  c2-2(^a  +  i?6  +  C'c), 

or. 


Aa  +  Bb  +  Cc 


^2  _|_   ^2    _^    C2  a2    +    ^,2    -|,    c2 

2  "^  2 

{A-aY  -ir{B-bY  +  {C-cf^ 
2 


which  in  Equation  (28-4)  gives, 

2m(^2_f.^2_^(^r2)_2m(„2_^52_^c2)=2m[(.l-a)2  +  (i?-i)2+(C-c)2], 

and   making 

A^  +  £-  +  C"  =  F2, 

«2     -f     62     _j_     g2      _     „2^ 

2  mV^  -  2  m  u^  =  2  m  [  {A  -  af  +  {B  -  hf  +  {C  -  cf^  •  •  (285) 

whence  we  conclude,  that  the  difference  of  the  sums  of  the  living 
forces  before  the  collision,  and  at  the  instant  of  greatest  compression, 
is  equal  to  the  sum  of  the  living  forces  which  the  system  would  have, 
if  the  masses  moved  with  the  velocities  lost  and  gained  at  this  stage 
of  the   collision. 

Since  all  the  terms  of  the  preceding  equation  are  essentially 
positive,  it  follows  that  at  the  instant  of  nearest  approach  of  the 
impinging  bodies,  there  is   a   loss   of  living    force. 

If  the  impinging  masses  now  react  upon  each  other  in  a  way  to 
cause  them  to  be  thrown  asunder,  and  A',  B',  C\  &c.,  denote  the 
components  of  the  actual  velocities,  in  the  direction  of  the  axes,  at 
the  instant  of  separation,  then  will  the  components  of  the  velocities 
lost  and  gained   while   the  separation   is  taking  place,  be 

a  —  A',     b  —  B',     c  —  C",    &c.,  &c. ; 

and   Equation   (283)  will   become 

2  m  {a  —  A')  a  +  I:  m  {b  —  B')  b  -\-  ^  m  (c  —  C')c  =  0, 
or, 

2  m  (a2  +  i2  4-  c2)  -  2  m  {A'  a  +  B' b  +   C  c)  =  0  ; 


208  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


and   eliminating  A' a  -\-  B' b  -{■  C  c,  by  means  of  the    identical  equa- 
tion, 

a2  +  ^.2  +  c2  +  ^'2  ^  ^/2 
-j- C' -  2  {A' a -]- B' b  +  C  c), 
we   obtain, 


(a  -  A'Y  +{b-  By  +  (c  -  c'f  =  I 


{a-A'Y^ 
2?n  (a2  +  P  +  c^)  -Urn  {A''-+B''-+  C'^)  =  -  2  7?^  <(    +  (6  -  ^')2  J. , 


/ 
and  making 


+  {c-  cy 


A'2   ^   iJ'2   4.    (7'2   ^    ■pr'2^ 

2  m  1*2  _  2  m  F'2  =  —  2  m  [  (a  -  ^l')2  +  (i  -  i?')2  +  (c  -  C")^]  •  •  (286) 

All  the  terms  of  this  equation  being  essentially  positive,  it  fol- 
lows, from  the  sign  of  the  second  member,  that  during  the  reaction 
of  the  bodies  by  which  they  are  separated,  there  is  a  gain  of  living 
force. 

If  the  loss  and  gain  of  velocities  after,  be  the  same  as  before 
the  instant  of  greatest  compression,  .then  will  there  be  no  loss  or 
gain   of  living   force   by  the    collision. 

The   principles    of    Equations    (285)    and    (286)    find    an    important 
application  in  the  construction,  adjustment,  and  motion  of  machinery. 

SYSTEM   OF   THE   WORLD. 

§  207. — The  most  remarkable  system  of  bodies  of  which  we  have 
any  knowledge,  and  to  which  the  preceding  principles  have  a  direct 
application,  is  that  called  the  solar  system.  It  consists  of  the  Sioi, 
the  Planets,  of  which  the  earth  we  inhabit  is  one,  the  Satellites  of  the 
planets,  and  the  Comets.  These  bodies  are  of  great  dimensions,  are 
spheroidal  in  figure,  are  separated  by  distances  compared  to  which 
their  diameters  are  almost  insignificant,  and  the  mass  of  the  sun  is 
so  much  greater  than  that  of  the  sum  of  all  the  others  as  to  bring 
the  common  centre  of  inertia  of  the  whole  within  the  boundary 
of  its  own  volume. 

These  bodies  revolve  about  their  respective  centres  of  inertia, 
are    ever    shifting   their  relative  positions,  and  our  knowledge  of  them 


MECHANICS     OF    SOLIDS.  209 

is   the   result  of  computations   based   upon   data  derived  from   actual 
observation. 

Kepler  found ; 

I.  That  tlve  areas  swept  over  by  the  radius  vector  of  each  planet 
about  the  sun,  t»  the  same  orbit,  are  proportional  to  the  times  of  de- 
scribing them. 

II.  That  the  j'jfo^ie/s  move  in  ellipses,  each  having  one  of  its  foci  in 
the  smi's  centre. 

III.  That  the  squares  of  the  periodic  times  of  the  planets  about  the 
sun  are  proportional  to  the  cubes  of  their  mean  distances  from  that 
body. 

These  are  called  the  laws  of  Kepler,  and  lead  directly  to  a 
knowledge  of  the  nature  of  the  forces  which  uphold  the  planetary 
system. 

§208. — The  first  law  shows,  §157,  that  the  centripetal  forces 
which  keep  the  planets  in  their  orbits,  are  all  directed  to  the  sun's 
centre ;  and   that   the    sun    is,  therefore,  the  centre  of  the  system. 

The  second  law  shows,  §  164,  that  the  law  of  the  centripetal 
force   is    that   of  the   inverse    square   of  the    distance. 

Denoting  by  T,  the  periodic  time  of  any  one  planet ;  by  a  and  b, 
the   semi-axes  of  its   orbit,  we  have.  Equation  (198), 

2  area  of  ellipse  _  2'n'ab  ^ 
~  27"  "~  "27""  ' 

and  substituting  the  values  of  i  and  c.  Equations  (212)  and  (211)' 


n  3 

^  ^  2*.a2 


yT 


whence, 


and  for  another  planet 


T^        4*2 


77  -IT 

14 


210  ELEMENTS     OF     ANALYTICAL    MECHANICS, 

but   by    Kepler's   third  law, 

and   therefore 


Avhence  Ave  conclude  that  not  only  is  the  laio  of  the  force  the 
same  for  all  the  planets,  but  the  absolute  force  is  the  same;  and  that 
the  same  cause  acts  upon  all  the  planets ;  and  that  if  the  planets 
were  at  the  same  distance  from  the  sun,  the  unit  of  mass  of  each 
would    experience   the    same   intensity  of  attraction. 

From  these  consequences  of  the  laws  of  Kepler,  it  is  inferred  that 
the  particles  of  the  sun  attract  those  of  the  planets,  and  vice  versa, 
with  a  force  varying  directly  as  the  mass  of  the  attracting  particle, 
and  inversely  as  the  square  of  the  distance.  And  from  the  experi- 
ments of  Dr.  Maskelyne,  who  found  by  observations  on  the  fixed 
stars,  that  the  mountain  Shehallien  in  Scotland  drew  the  plumb  line 
sensibly  from  its  vertical  position ;  and  also  from  the  experiments 
of  Cavendish  and  Baily  upon  leaden  and  other  balls,  it  is  inferred 
that  this  power  of  attraction  resides  in  every  particle  of  matter, 
wherever  found,  and  that  it  is  exerted  under  all  circumstances  without 
the  possibility  of  being  intercepted.  It  is  therefore  concluded  that 
matter  is  endowed  with  a  general  gravitating  principle  by  which  every 
particle  attracts  every  other  particle,  and  according  to  the  law  above 
mentioned. 

But,  according  to  this  principle,  not  only  does  the  sun  attract  the 
planets,  but  the  planets  attract  the  sun  and  one  another.  Either 
Kepler's  laws  cannot,  therefore,  be  rigorously  true,  or  universal  gravi- 
tation is  not  a  Principle  of  Nature.  Now  in  point  of  fact,  observa- 
tions of  fiir  greater  nicety  than  those  of  Kepler,  prove  that  his  laM"s 
are  not  accurately  true,  though  they  differ  but  slightly  from  reality ; 
a  circumstance  arising  entirely  from  the  fact  of  the  great  mass  of 
the  sun  as  compared  with  the  sum  of  the  masses  of  all  the  planets. 
Were  there  but  a  single  body  in  existence  besides  the  sun,  it  would 
describe   accurately   an    elliptical,  parabolic  or  hyperbolic   orbit   about 


MECHANICS     OF     SOLIDS.  211 

the  common  centre  of  inertia,  depending  upon  its  living  force  and 
the  sun's  attraction.  A  third  body  would  derange  this  motion  and 
cause  a  departure  from  the  regular  path,  and  the  degree  of  the 
disturbance  would  depend  upon  the  mass,  distance  and  direction  of 
the  disturbing  body  as  compared  -with  those  of  the  sun.  The  same 
remark  would  apply  to  a  fourth,  fifth,  and  to  any  number  of  addi- 
tional bodies.  These  disturbances,  by  which  any  one  body  of  the 
system  is  made  to  depart  from  the  simple  path  due  to  the  sun's 
action  alone,  and  which  are  caused  by  the  combined  action  of  all 
the  others,  are  called  ^;er^?<ria//o«5.  These  have  been  computed,  and 
the  complete  harmony  which  is  found  to  subsist  between  the  numeri- 
cal results  deduced  from  theory  and  observation,  is  the  strongest 
possible  evidence  in  support  of  the  Law  of  Universal  Gravitation. 

If  the  principal  plane  of  the  solar  system,  as  determined  at 
different  and  remote  periods,  be  found  to  have  undergone  no  change, 
this  will  show  that  the  system  is  uninfluenced  by  the  action  of  the 
fixed  stars  and  other  distant  bodies,  and  its  centre  of  inertia  will, 
§  198,  either  be  at  rest  or  be  moving  uniformly  through  space  in 
a  right  line ;  but  if  the  principal  plane  be  found  to  change  its 
place,  it  will  be  a  sign  that  the  system  is  in  motion,  and  that  its 
centre  of  inertia  is  describing  a  curvilinear  path  about  some  distant 
centre. 

EMPACT   OF  BODIES. 

§209. — When  a  body  in  motion  comes  into  collision  with  another, 
either  at   rest   or   in   motion,  an  im2)act  is   said   to    arise. 

The  action  and  reaction  which  take  place  between  two  bodies, 
when  pressed  together,  are  exerted  along  the  same  Tight  line,  per- 
pendicular to  the  surfaces  of  both,  at  their  common  point  of  contact. 
This  arises  from  the  symmetrical  disposition  of  the  molecular  springs 
about   this  line. 

When  the  motions  of  the  centres  of  inertia  of  the  two  bodies 
are  ^rarotZ/e/  to  this  normal  before  collision,  the  impact  is  said  to  be 
divff. 

When    this    normal  passes    through   the    centres  of  inertia  of  both 


212 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


bodies,    and   the   motions    of   these    centres   are    along    that   line,    the 

impact     is    said    to     be     direct    and 

central. 

When  the  motion  of  the  centime 
of  inertia  of  one  of  the  bodies  is 
along  the  common  normal,  and  the 
normal  does  not  pass  through  the 
centre  of  inertia  of  the  other,  the 
impact  is  said  to  be  direct  and 
eccentric. 

When  the  path  described  by  the 
centre  of  inertia  of  one  of  the  bodies, 
makes  an  angle  with  this  normal, 
the  impact  is  said  to  be  oblique. 

When  two  bodies  come  into  col- 
lision, each  will  experience  a  pres- 
sure from  the  reaction  of  the  other  ;  and  as  all  bodies  are  more  or 
less  compressible,  this  pressure  will  produce  a  change  in  the  figure 
of  both ;  the  change  of  figure  will  increase  till  the  instant  the  bodies 
cease  to  approach  each  other,  when  it  will  have  attained  its  maximum. 
The  molecular  spring  of  each  will  now  act  to  restore  the  former 
figures,  the    bodies  will  repel  each    other,  and  finally  separate. 

Three  periods  must,  therefore,  be  distinguished,  viz.  :  1st.,  that 
occupied  by  the  process  of  compression  ;  2d.,  that  during  which  the 
greatest  compression  exists ;  3d.,  that  occupied  by  the  process,  as 
far  as  it  extends,  of  restoring  the  figures.  The  force  of  restitution 
must  also  be  distinguished  from  tlie  force  of  distortion;  the  latter 
denoting  the  reciprocal  action  exerted  between  the  bodies  in  the 
first,  and  the  former  in   the   third   period. 

The  greater  or  less  capacity  of  the  molecular  springs  of  a  body 
to  restore  to  it  the  figure  of  which  it  has  been  deprived  by  the 
application  of  some  extraneous  force  when  the  latter  ceases  to  act, 
is  called   its  elasticity. 

The  ratio  of  the  force  of  distortion  to  that  of  restitution,  is  the 
measure  of  a  body's  elasticity.  This  ratio  is  sometimes  called  the 
co-efficient  of  elasticity.      When    these   two   forces   are   equal,  the   ratio 


MECHANICS    OF    SOLIDS. 


213 


is  unity,  and  the  body  is  said  to  be  'perfectly  elastic ;  when  the 
ratio  is  zero,  the  body  is  said  to  be  non-elastic.  There  are  no  bodies 
that  satisfy  these  extreme  conditions,  all  being  more  or  less  elastic, 
but   none  perfectly  so. 

Let  the  two  bodies  AB  and  A' B',  the  former  moving  along  the 
line  J£  T,  and  the  latter  along 
H'  T',  come  into  collision  at  the 
point  0.  Through  0,  draw 
the  common  normal  JVL.  De- 
note the  angle  H  G  N  by  9, 
and  H'  EN  by  9' — these  being 
the  angles  which  the  directions 
of  the  two  motions  make  with 
the  normal.  Also  denote  the 
velocity  and  mass  of  the  body 
^4^  by  V  and  M  respectively,  and  the  velocity  and  mass  of  A'  B' 
by    V  and  M'. 

The  components  of  the  quantity  of  motion  of  the  two  bodies  in 
the  direction  of  the  normal  and  of  the  perpendicular  to  the  normal, 
will   be 

M  V  cos  9,     M'  V  cos  9'     and     M  V  sin  9,     M'  V  sin  9'. 

The  former  of  these  components  Avill  alone  be  involved  in  the 
impact ;  for  if  the  bodies  were  only  animated  by  the  latter,  they 
would  not  collide,  but  would  simply  move  the  one  by  the  other. 
For  simplicity,  let  the  body  A  B  ha  spherical ;  the  normal  will 
pass    through   its    centre  of  inertia. 

Denote  by  «,  the  velocity  of  the  body  A  B  \n  the  direction  of 
the  normal  at  the  instant  of  greatest  compression,  and  by  u'  the 
velocity  of  the  body  A'  B'  at  the  same  instant  in  the  same  direction. 
Then   will 

V  cos  9  —  «,     and      V  cos  9'  —  u'     •      •     •     (287) 

be  the  velocities  lost  and  gained  in   the  direction  of  the  normal,  and 

M{Vcos(p  —  u),     and     J/' ( 7' cos  9'  —  u')    •  •  •  (288) 


214         ELEMENTS     OF    ANALYTICAL    MECHANICS. 

be  the  forces  lost  and  gained  at  the  instant  of  greatest  compression ; 
and   hence, 

M{Vcos  <p  -  u)  J-  M'  (F'coscp'  -  7/)  =  0;  •     •     (289) 

and  denoting  the  angular  velocity  of  the  body  A' B'  by  F/ ,  the 
distance  G'  D  from  the  centre  of  inertia  of  A'  B'  to  the  normal 
by  e,  and  the  principal  radius  of  gyration  of  A'  B\  with  reference 
to    the   instantaneous  axis  by  k^ ,  then  will 

_  Jf(Fcos(p  -  ii).e  ^  (Fcos(p-«)e  ,^^^. 

^'    -  Mk^  k^  ^       ' 

and  since  the  velocity  u  must  be  equal  to  that  of  the  point  B  at 
the   end   of  the   lever   arm  e,  we   have 

u  =  u'  ^  e.V; (291) 

Substituting  the  values  of  u  and  w'  from  this  equation  successively 
in   Equation  (289),  we  find 

if  F  cos  ^  +  M'  V  cos  9'  4-  ^'  ^  y,' 


M  +  M' 
i¥  Feos  0  +  M'  V  cos  ^'  —  M  e  V ,' 


(292) 
(293) 


After  the  instant  of  greatest  compression,  the  molecular  springs 
of  the  bodies  will  be  exerted  to  restore  the  original  figures,  and 
if  c  denote  the  co-efficient  of  elasticity,  then  will  the  velocities  lost 
\i^  AB  and  gained  by  A'  B'  during  the  process  of  restitution  be, 
respectively, 

c  (  F  cos  9  —  «)     and     c  (F'  cos  9'  —  u') ; 

and   the   entire   loss   of  A  B,  and  gain  of  A'  B',  will  be,  respectively, 

Fcos9 -M  +  c(F cos 9 —  «),    and     V cos:?' —u' +  c{V' eoscp'— u'). 

Also    the    gain    of    angular  velocity    of  the    body  A'  B\  during    the 
process  of  restitution,  will  be 

(  Fcos  (p  —  «) .  c 


MECHANICS     OF    SOLIDS.  215 

and  the  whole  anguUir  velocity  produced  by  the  impact  and  denoted 
by    V^,  will    be   given    by    the    equation, 

Denoting  the  velocities  of  A  B  and  A'  B\  after  the  collision  by 
V  and  v\  and  the  angles  whieh  the  directions  of  these  velocities 
make  with  the  normal  l)y  (3  and  d',  respectively,  then  will 

v  cos  (3  z=  P^cos(p  —  Fcos(p  +  u  —  c  (Fcos9—  xl)  —  (\  -\-  c)  u  ~  c  Tcosp, 

v'cos^'^  F'cos9—  F'cos(p'  +  «'  — c(F' cos 9'  — m')  =  (1+c)u'  — cP"  cos  9', 

and  replacing  the  values  of  u  and  u\  as  given  by  Equations  (292) 
and  (293), 

,     ,,       ,  JjTF cos  9+ jif' F' cos 9'+ iTe  F/ 
t;cosd..:(l  +  c)  •      ^     M+M'  ^-cFcos9,   (295) 

„     ,,       ,  J/Fcos9+l/' F'cos9'-J/e  F/  ,,,  ,,       ^ 

v'cos(3'  =  (l+c)  1^:^-^ -^ ^  —c  F'cos9'   29G) 

Moreover,  because  the  efTects  of  the  impact  arising  from  the  corapo 
nents  of  the  quantities  of  motion  in  the  direction  of  the  normal  will 
be  wholly  in  that  direction,  the  components  of  the  quantities  of 
motion  before  and  after  the  impact  at  right  angles  to  the  normal  will 
be  the  same,  and  hence 

V  sin  ^  =  Fsin  9, (297) 

v'^na'  =  F'sin9' (298) 

Squaring  Equations  (295)  and  (297)  and  adding ;  also  Equations 
(296)  and  (298)  and  adding,  we  find  after  taking  square  root,  and 
reducing  by  the  relations 

cos2  a  +  sin2  a  =  1  ;    cos2  0'  -f-  sin2  d'  =  1  ; 

r         , M Fcos9 + M'  F'cos9' + M'JV.'       ~       -,,  ,   t-o  •  ^      /-,an\ 
^^V  [(1+0 M  +  J     ^-cT'cos9P+  F"sm29.  (299) 


/r , ,   ,    .M Fcos9  +  M'  F'cos9 ' — Me  V '  .       , /oaa\ 

v'  =y^  [(1  +  c) M+  M'    ■  ^^^'''^  ^  F'2sm29'.(300) 


216         ELEMENTS     OF     ANALYTICAL    MECHANICS. 

Dividing  Equation  (297)    by  Equation  (295),  and    Equation  (298)  by 
Equation  (29G),  we  have, 

^  •  ^^^^  ^ 
^^"^  ^  =  ":           ,  M  Vcoscp  +  M'  F'cos  ^'  +  M' e  V/         ;;  '^^^^^ 

(I  +  c) ^—^ — TT ^rn-^ — ' —  c  Fcos  (D 

F'.sincp' 

^^"^'^^           ,M  Fcosa.  +  M'  V  C0S9'  -  Me  V/         ~         ;*^"     ^ 
(1  +  ^) ^ ^^Tm' --cF'cos,' 

Equations  (290)  and  (292),  will  give  the  values  of  u  and  F/,  in 
known  terras,  and  these  in  Equations  (294),  (295)  and  (296)  will 
give  the  values  of  F^,  v,  and  v',  and  all  the  circumstances  of  the 
collision  will  be  known. 

g  210. — If  the  bodies  be  both  spherical,  then  will  e  =  0,  and  Equa- 
tion (294)  gives  F,  =  0 ;  and  Equations  (299)  and  (300),  (301)  and 
(302),  become 


MVcOSCp  +  M'V   COS(p'  ,    iro    •    2  /QAQ\ 

— cFcostp  -+ F^sm^m  •  •  •  (303) 

M  +  M'  ^-'  ^  ^      ' 


.=^[(l+c) 

77        ,  MVcoscp  +  M'  V  cos  cp'       Z  .       ,       ., 

v'=-^[(l+c) -^-M' cF'cos9']2+F'2sm2<p'  .  .  (, 


!04) 


t''^^^  =  ; ,J/Fcos^+.¥'rcos9'         '  '  '  ^"^^^ 

(1+0 1^-,^, cFcos, 

tan ^  = :wv^^^WIrr^^'        ~       ;  ■  ■  ^^^^ 

(1+^) M^ll' ^-^>^'cos,' 

The  Equations  (303)  and  (304)  will  make  known  the  velocities, 
and  (305)  and  (306)  the  directions  in  which  the  bodies  will  move, 
after  the  impact. 

Now,  suppose  the  body  A'  B'  at  rest,  and  its  mass  so  great  that 

the  mass   of    AB    is    insignificant    in    comparison,    then    Avill    F'   be 

M 
zero,  M'  may  be  written  for  if  +  M'  and  -ry-  will  be  a  fraction  so 


MECHANICS     OF     SOLIDS. 


217 


small  that   all    the    terms    into    which   it    enters  as    a    factor    may    be 
neglected,  and  Equation  (303)    becomes 


V  =  V  -y/  c'^  cos^  9  +  sin^  9  ; 


and   Equation   (305), 


tan  6  =  — 


tan  (p 


(307) 


The  tangent  of  6  being  negative,  shows  that  the  angle  NHX, 
which  the  direction  of  AB's  motion 
makes  with  the  normal  iViV'  after  the 
impact,  is  greater  than  90  degrees;  in 
other  words,  that  the  body  AB  is 
driven  back  or  reflected  from  A' B'. 
This  explains  why  it  is  that  a  cannon- 
ball,  stone,  or  other  body  thrown  ob- 
liquely against  the  surface  of  the  earth, 
will  rebound  several  times  before  it 
comes   to    rest. 

If  the  bodies  be  non-elastic,  or,  which  is  the  same  thing,  if  c  be 
zero,  the  tangent  of  (3  becomes  infinite ;  that  is  to  say,  the  body 
A  B  will  move  along  the  tangent  plane,  or  if  the  body  A'  B'  were 
reduced  at  the  place  of  impact  to  a  smooth  plane,  the  body  xi  B 
would   move   along   this   plane. 

If  the  body  were  perfectly  elastic,  or  if  c  were  equal  to  unity, 
which    expresses  this   condition,  then    would    Equation  (30T)  become 


tan  d  =  —  tan  9 


(308) 


which  means  that  the  angle  NHF  =  E H N"  becomes  equal  to 
KHN'.  The  angle  EHN'  is  called  the  angle  of  incidence,  the 
angle  KHN',  commonly,  the  angle  of  reflection.  Whence  we  see, 
that  when  a  perfectly  elastic  body  is  thrown  against  a  smooth,  hard, 
and  fixed  plane,  the  angle  of  incidence  will  be  equal  to  the  angle 
of  reflection. 

If  the  angles  9  and  9'  be   zero,  then  will    cos  9  =  1,    cos  9'  =  1, 


218  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

sill  9  =  0,    sin  (p'  —  0;    the    impact   will    be    direct   and    central,  and 
Equations  (303)  and  (304)  become 

and  passing  to  the  limits,  non-elasticity  on  the  one   hand  and  perfect 
elasticity  on  the  other,  we  have  in  the  first  case,  c  =  0,  and 

M  V  +  M  V 
''  =  -M^M- W 

MV  +  WV 

and   in    the    second,  c  =  1,  consequently, 

^MV  -\-  M'  V 
^-^-If+iT^-^ (311) 


CONSTEAESTED   MOTIOIf. 

§211. — Thus  far  we  have  only  discussed  the  subject  o^  free  motion. 
We  now  come  to  constrained  motion. 

Motion  is  said  to  be  constrained  when  by  the  interposition  of 
some  rigid  surface  or  curve,  or  by  connection  with  some  one  or 
more  fixed  points,  a  body  is  compelled  to  pursue  a  path  different 
from    that   indicated   by    the   forces   which  impart   motion. 

§  212. — The  centre  of  inertia  of  a  body  may  be  made  to  con- 
tinue on  a  given  surface,  by  causing  it  to  slide  or  roll  upon  some 
other  rigid  surface. 

§213. — We  have  seen,  §128,  that  the  motion  of  ♦translation  of 
the    centre   of  inertia,    and    of  rotation    about   that   point,  are    wholly 


MECHANICS     OF     SOLIDS. 


219 


independent  of  one  another,  and  the  generality  of  any  discussion 
relating  to  the  former  -will  not,  therefore,  be  affected  by  making, 
in    Equation  (40), 

Sep  =  0;     5-^  =  0;     6zi  =  0; 
■\vhieh  will  reduce  that  equation  to 

(2  Pcos  a  —  -— -  •  2  m)  o  x^ 

+  (2Pcos/3-  -^.2m)oy,    V  =  0. 

d'^z 
+  (2Pcos7  -—.y.m)oz, 

Making 

2  m  —  M;     2  P  cos  a  =  X-     2  Pcos  /3  =   F;     2  P  cos  7  =  Z; 
and  omitting  the  subscript  accents,  we  jnay  write  f 

(52=0.(313) 

inertia,  and 
surface    of 


.     .     (314) 
morality    of 
Equation    (313),  is    restricted    to  the  conditions    imposed    by  this  cir- 
cumstance. 

Supposing  the  variables  x  ij  z,  in  the  above  equations,  to  receive 
the  increments  or  decrdnents  S  x,  (5y,  S  z,  respectively,  we  have,  from 
the  principles  of  the  calculus, 


dL  dL  dL 

-,-'6x  +  -^  -Sy  +   ---dz  =  0. 

ax  ay  dz 


(315) 


Multiplying  by  an  indeterminate   quantity  X,  and  adding   the   product 
to  Equation  (313),  there  will  result 


(X-...-  +  X" 


dL 


7)-' 


/^^         ^r    ("~'f     ,    .      dL\   . 


+ 


(^-^■?^  +  -")-J 


21S  ELEMENTS     OF     AXALYTICAL    MECHANICS. 

sill  (p  =  0,    sill  9'  =  0  ;    the    impact   will   be   direct   and   central,  and 
Equations  (303)  and  (304)  become 

.^  ^    ,  J/F+  MY'         _ 
^  =  ('  +  ^)       M^M       -  ^^' 

and  jiassing  to  the  limits,  non-elasticity  on   the   one   hand  and  perfect 
elasticity  on  the  other,  we  have  in  the  first  case,  c  =  0,  and 

M  V  ^  M  V  ,       , 


and   in    the 


'  J^    ^     ^-^     ^     "^^"^ 

COXSTEAINED   MOTIO^T. 

§211. — Thus  far  we  have  only  discussed  the  subject  oi  free  motion. 
We  now  come  to  constrained  motion. 

Motion  is  said  to  be  constrained  when  by  the  interposition  of 
some  rigid  surface  or  curve,  or  by  connection  with  some  one  or 
more  fixed  points,  a  body  is  compelled  to  pursue  a  path  different 
from    that   indicated   by    the   forces   which  impart   motion, 

§  212. — The   centre   of  inertia   of  a  body    may   be   made   to    con-    • 
tinue   on  a   given   surface,   by  causing  it  to   slide   or  roll    upon  some 
other  rigid  surface. 

§  213. — We  have  seen,  §  128,  that  the  motion  of  ♦translation  of 
the   centre    of  inertia,    and   of  rotation    about   that   point,  are   wholly 


MECHANICS     OF    SOLIDS. 


219 


independent  of  one  another,  and  the  generality  of  any  discussion 
relating  to  the  former  will  not,  therefore,  be  affected  by  making, 
in    Equation  (40), 

o<p  -  0;     S-^  =  0;     duJ  =  0; 
■which  will  reduce  that  equation  to 

(2  F  cos  a  —  — -  •  2  m)  S  x^ 


+  (2Pcos/3-  -^.2TO)oy, 


+  (2Peos7  — 


I)  0  z, 


^  =  0. 


Making 

2  m  =  M;     IP  cos  a  =  X;     2  Pcos  /3  =   F;     IP  cos  y  =  Z; 
and  omitting  the  subscript  accents,  we  may  write  < 

(.V-i/.^)  0.  +  (r-J/.'-g)  oy  +  (Z-M-'-^)  fe=0.(313) 

Now,  assuming  the  movable  origin  at  the  centre  of  inertia,  and 
supposing  this  latter  point  constrained  to  move  on  the  surface  of 
which  the  equation   is 

L  =z  F{xyz)  =  0, (314) 

the  virtual  velocity  must  lie  in  this  surface,  and  the  generality  of 
Equation  (313),  is  restricted  to  the  conditions  imposed  by  this  cir- 
cumstance. 

Supposing  the  variables  x  y  z,  in  the  above  equations,  to  receive 
the  increments  or  decri^nents  S  x,  (Jy,  S  z,  respectively,  we  have,  from 
the  principles  of  the  calculus. 


dL    ^  dL     .  dL     ^ 

r-'Ox  -\-  -y-  'dy  +  ----02  =  0. 
dx  d  V  dz 


(315) 


Multiplying  by  an  indeterminate   quantity  X,  and  adding    the    product 
to  Equation  (313),  there  will  result 


/^        ,^    d^-x        ,      dL\   ^      ^ 
\  dt^  dx/ 

/^^         ,^    (Pil  d L\  ^ 


=  0. 


220 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


The  quantity  X,  being  entirely  arbitrary,  let  its  value  be  such  as  to 
reduce  the  coefficient  of  one  of  the  variables  S  x,  o  y,  S  z,  say  that  of 
5x,  to  zero;  and  there  will  result 


,,  d^x       ^     dL 
jf  _  3/  ^  +  X  .  — -  z=  0, 

at-  ax 


(316) 


and 


Now  in  Equation  (315),  oy  and  5z  may  be  assumed  arbitrarily,  and 
5x  will  result;  hence  o  y  and  5z  in  Equation  (317)  may  be  regarded 
as  independent  of  each  other,  and  by  the  princijile  of  indeterminate 
coefficients, 


\ 


^^ 


d  t"  d  y 


^^   d^z     ^    ^     dL 


(318) 


and   eliminating  X  by  means  of  Equation  (316),  we  find, 

(r-3/.!^!).^-(x-3/.ff).l'^  =  o. 

\  dt~y       dx         V  dt^y      dy 

\  d  t^y      dy         \  dt-  ^      dz  '  j 


(319) 


which,  with   the  equation  of  the  surface,  will  determine  the  place  of 
the    centre  of  inertia  at  the  end  of  a  given  time. 


MOTION    ON    A   CURVE   OF   DOUBLE   CURYATUKE. 

8  214. — If  the  centre  of  inertia  be  constrained  to  move  upon 
two  surfaces  at  the  same  time,  or,  which  is  the  same  thing,  upon 
a  ciirve  of  double   curvature  resulting  from  their  intersection,  take 


L  =  F{xyz)  =  0, 
H=F'{xyz)  =  0 


■\ 


(320) 


MECHANICS     OF     SOLIDS. 


221 


from  which,  by  the  jiroccss  of  differentiating  and  replacing  dx,  dy,  dz, 
by  the   projections  of  the   virtual    velocity, 


dL  ^  dL               dL 

a  X  ay                 a  z 

dll  .  ^dH  .     ^dll   . 

ax  ay                 a  z 


(321) 


(322) 


Multiplying  the  first  of  these  by  X,  and  the  second  by  X',  adding  the 
products  to  Equation  (313),  and  collecting  the  coefficients  of  S  x,  Sy, 
and  6  2,  we  have 


(x  -  M. 


+ 


(>'- 


M. 


+  (z-J/.::^  +  X 


d-^x 
df 

+ 

X 

dL 
dx 

+ 

X' 

d  II\  . 
'-dx)' 

d-  y 
de 

+ 

X 

dL 
dy 

+ 

X' 

dH\   . 

■iyr 

d^z 

dL 

dH\   . 

dt^ 


dL  ,    d H\   . 

dz  d  z  y 


^=0     •  (323) 


Now  the  coefficients  of  two  of  the*  three  variables  ^x^  5y,  and  8  z, 
say  those  of  5x  and  By,  may  be  made  equal  to  zero  by  assignuig 
proper  values  for  that  purpose  to  the  indeterminate  quantities  X  and 
X',  in  w  hich  case,  since  ^  s  is  not  equal  to  zero,  its  coefficients  must 
also  be  equal  to  zero ;    whence 


a  r  d  X  dx 


d  t"  dy  dy 


.^,    Z 


M- 


d'^ 


dH 

dy 
dH 


dfi  d  z  d  z 


(324) 


and  eliminating  X  and  X',  there  will  result 


/t,       ,r   d'^x\     {d  L     dH        dL     dH\    ^ 

KX  —  M-  —r-r  )  •  i  —r~  •  -, r—  •  -7 —  ) 

\  a^^/      \  dz      d  y  dy      d  z  / 


y 

H 
dJ 


/,^       ^rd-v\     {d  L     dH         dL     d  H\  „     ,„.^_. 

^\  di-y      \d.T      dz  dz      dx/      ^  ^       ' 

'd  L     d  H 


d  z      dx 
dL     dH' 


/  _  d'z\      ML     dH         dL     dH\ 

\  dt-y      ^dy       dx  dx       dy/ 


222 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


which,  with  the    equations  of  the    surfaces,  is   sufficient    to    determine 
the  co-ordinates   of  the    centre   of  inertia  when  the   time   is   given. 


§  215. — If    the    given    surfaces    be    the    projecting   cylinders   of    a 
curve  of  double  curvature,  then   will    Equations    (320)    become 


(326) 


And  because  L  is  now  independent  of  y,  and   H  is  independent  of  x^ 


we  have 


ay  ax 


which  reduce  Equations  (324)  to 


dv-  dx 

-.^    d^  V  ,    d  H 

Y  -  M'-f^  +X'.— -  =  0; 
dt~  ay 

,^    d^z         ^      dL  ,    dH       ^ 


(327) 


and  Equation    (32.5)  to 


/..  .r    d-x\  dL  dff^ 

V  d  t-  y  dz  dy 

/^^  ,,    d- y\  dL  dH 

\  d  t~  ^  d  X  d  z 

\  dt"/ 


dL    dH 

dx     dy 


(328) 


This,    with    the   equations   of    the   curve,    Avill    give   the   place   of   the 
centre    of  inertia   at   the    end    of  a   given   time. 

§216. — If  the  curve  be  plane,  the  co-ordinate  plane  xz,  may  be 
assumed  to  coincide  with  that  of  the  curve ;  in  which  case  the 
second  of  Equations  (327),  becomes  independent  of  y,  that  varia- 
ble  reducing  to    zero,    and 

1     TT 

d-^y  =  0,     and     — —  =  0 ; 
-^  dy  ' 


MECHANICS     OF    SOLIDS. 

hence  Equations    (327),  bcome 

d'~  X  d  L         - 

X-  if.--  +  X.--  =  0; 
dt-  d  X 

F-0; 

,,    d-'z     ,    ,     dL    ^    ^,  dH 
dl^  dz  dz 


223 


(329) 


and  because  the   foctor 


d"^  V 


dH 

Equation   (328)   becomes,   on   dividing  out   the   common  factor   — , 

(X  -  M-  ^)  .  1^  -  (Z  -  J/,  ^r)  •  ^  =  0.  .  (330) 
\  di-J      dz         V  dt'J      dx 

§  217.— By    transposing  the  terms  involving  X,  in  Equations  (316) 
and    (318)  and   squaring   we   have 


4  ("J+ (^7)^+ 0> 


The  second  member  of  this  equation  is.  Equation  (50),  the  square  of 
the  intensity  of  the  resultant  of  the  extraneous  forces  and  the  forces 
of  inertia.     Denoting   this  resultant   by  i\^,  we  may  Avrite 


and  dividing  each  of  the  equations 
dL 


X 

X- 
X 


i-J 


d_L_  _        / 
Tx   ~  ~  \ 

dij  \  dt~/ 

dz  V  di~y  ' 


224  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

obtained    by    the   transposition  just   referred   to,    by    Equation    (331), 
we  find, 

dL 
d  X 


dL 
dy 


s/m^(^i\)'-m 


dLS" 
)  +  K 

y 

dj^ 

dz 


X  - 

M- 

d"^  X 

dt 

N 

Y  - 

M- 

d'^  y 
d  t" 

N 

Z- 

-M 

d-^  z 

/W^  (iW-  (-i)' 


(332) 


The  second  members  are  the  cosines  of  the  angles  which  the 
resultant  of  all  the  forces  including  those  of  inertia,  makes  with  the 
,ic.^f  7"i  axes;  the  first  members  are  the  cosines  of  the  angles  which  the 
normal  to  the  surface  at  the  body's  place  makes  with  the  same  axes. 
These  being  equal,  with  contrary  signs,  it  follows  not  only  that  the 
forces  whose  intensities  are 


v('^y+(^y+(^x--' 

are  equal,  but  that  they  are  both  normal  to  the  surface,  and  act  in 
opposite  directions.  The  second  is  the  direct  action  upon  the  surface; 
the  first  is  the  reaction  of  the  surface. 

§  218. — If  the  last  terms  in  Equations  (3IG)  and    (318)    be    multi- 
plied and  divided  by 

and  the  angles  which  the  normal  resistance  of  the  surface  makes  with 


MECHANICS    OF    SOLIDS.  225 

the   axes   x,  y,  z,    respectively,    be   denoted   by   d',  6"  and  d"\   those 
equations  Avill  take   the   form 

cPx 

X  —  M  •  — -  -f  N'  cos  ^'  =  0 ; 


F-  i/.^  +  iVT.cosd"  =0; 
Z  -  J/ •  ---  +  iV^.  cos  ^"'  =  0. 


(333) 


§219, — To  impose  the  condition,  therefore,  that  a  body  in  motion 
shall  remain  on  a  rigid  surface,  is  equivalent  to  introducing  into 
the  system  an  additional  force,  which  shall  be  equal  and  directly 
opposed  to  the  pressure  upon  the  surface.  The  motion  may  then 
be  regarded  as  perfectly  free,  and  treated  accordingly.  The  same 
might  be  shown  from  Equations  (324)  to  be  equally  true  of  a 
rigid  curve,  but  the  principle  is  too  obvious  to  require  further 
elucidation. 

Equations  (333),  may,  therefore,  be  regarded  as  equally  appli- 
cable to  a  rigid  curve  of  any  curvature,  as  to  a  surface ;  the  nor- 
mal reaction  of  the  curve  being  denoted  by  JSf^  and  the  angles 
which   N  makes   with    the   axes   x,  y,  z,   by    ^',  &"   and    &'". 

§  220. — To  fmd  the  value  of  iV,  eliminate  d  t  from  Equations 
(333),  by  the   relation 

J_  _  JT 

dt    ~  ds' 

in   which   V  and  5  are  the  velocity  and   the  space ;   then  by  transpo- 
sition  these   equations   may   be   written 

iVT.cos^'  =  M-  V^'-rl  -  X; 
a  a^ 

N-COS6"  =  M-  F2.^  -  Y- 

N-C0S6'"  ^M-  F2.-^  -  Z. 
a  s^ 

15 


226 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


Squaring,    adding   and   reducing   by   the  relations 

i22  =  X2  +  F2  +  Z2, 
cos^d'  +  cos^6"  +  cos2^"'  _  1^ 


and   we   find 


^2    ^      , 


-  2  M'  F2 


[x.^  +  F.^l^  +  Z.^' 

L  0?  S2  (£§2  ^/g 

Multiplying  the  last   term   of  the    second  member   by 


R 


making 


X       <Px     ^     Y        cPy        Z 


R  P-rf^  +  ^•p-Tri  +  7?-p-^  =  <^«« 


7^2 


i£2   =    7^2  sin2  (p    4-    i22  cos2  (p^ 


in  which  <p  denotes  tlie  angle  made  by  the  radius  of  curvature 
and  the  direction  of  the  resultant  R,  and  p  the  radius  of  curvature, 
we  get 


N^  = 


^I'  ■  ^,  ■  [{d'  .vY  +  {cP  y)2  +  (c/2  ,)2j 


i/F2 


R  cos  (p  +  i22  siu2  (p  +  it2  cos2  (p. 


But  i2  sin  (p,  is  the  component  of  the  resultant  R,  in  the  direction 
of  the  tangent  to  the  curve,  and  is  only  opposed  by  the  inertia  of 
the   body.     Whence 

Rsm^=-=V^.—, 
and 


i22sin2<p  =   V*-^^; 


MECHANICS     OF    SOLIDS. 


227 


which  substituted  above  and  reducing  by  the   relation 
P  = 


V^2a;)2  +  (rf2y)2  +  {d'^zy  -  {d'sy 


we    have,    regarding   s   as    the    independent   variable,   in   which    case 
<r~s  =^  0, 


and  taking  square  root. 


74  J/  F2 

--  —  2  — — -  •  E  cos  9  -f-  i22  cos2  2, . 

pi  P 


iv  = ^  cos  (p. 


(334) 


The  first   term    of  the    second   member  is, 

§  167,    the    centrifugal    force    arising    from 

the  deflecting  action  of  the  curve,  and  the 

last  term  is  the  normal  component  of  the 

resultant   H.     As  the   equation   stands,  its 

signs  apply  to  the  case  in  which  the  body 

is  on   the   concave   side  of  the   curve,  and 

the   resultant   acts   from  the   curve.     The  angle  9,  must  be   measured 

from  the    radius    of  curvature,  or    that   radius   produced,  according  as 

the  body    is    on    the   concave   or    convex    side   of   the    curve.      When 

the    body   is    moving   on    the   convex    side   of    the    curve,   the  first 

term    of    the    second    member    must    change    its    sign    and    become 

negative. 

§221. — Writing  Equations  (333)  under  the  form 

d'^x 
M.  ~  =  X  +  iV^cosa', 
dt^  ' 

M-^  =  Y+  Ncos6'\ 

M-^=  Z  -\-  i\^cos5"': 
dt^  ^  ' 


multiplying  the  first  by  2dx,  the  second  by  2rfy,  the  third  by  2dz, 
adding  and  reducing  by  the  relation       ^''^re    dJL  =  /"fs^  ,,  ^  <?. 

dz 


c?  s  1  -—  •  cos  ^'  +  -^  •  cos  d' 
M  5  ds 


+  -— • cos  d 
ds 


'•')-  =  o, 


228         ELEMENTS     OF     ANALYTICAL    MECHANICS. 

the  first    member   being   the   cosme   of  the   angle   made  by    the   nor- 
mal   and   tangent   to    the   curve,  we  have 

(2dx.d^x  +  2dy.d^V  +  2d~..d^^z\  ^^(^^d.+  Tdy  +  Zdz); 
\  d  f  ^ 

integrating  and  reducing  by 

dx'^dyl^rd^ 

we  find 

MV^  -^1  f{Xdx  +  Ydy  +  Zdz)  +  C.     '     '     (335) 

This  beinc  independent  of  the  reaction  of  the  curve,  it  can  have  no 
effect  upon   the   velocity. 

If  the   incessant   forces   be   zero,  then  will 

A^  =  0  ;     r  =  0  ;    and    Z  =  0  ; 

and 

7^  -  £.  . 
i/' 

that  is,  a  body  moving  upon  a  rigid  surflicc  or  curve,  and  not  acted 
upon  by  incessant  forces,  will  preserve  its  velocity  constant,  and  the 
motion  will   be   uniform. 

We  also  recognize,  in  Equation  (335),  the  general  theorem  of 
the  livino'  force  and  quantity  of  work  ;  and  from  which,  as  before, 
it  appears  that  the  velocity  is  wholly  independent  of  the  path  de- 
scribed. 

Example  1. — Let  the  body  be  required  to  move  upon  the  interior 
surface  of  a  spherical  bowl,  under  the  action  of  its  own  weight.  In 
this   case, 

Z  =  a;2  +  y2  +  ^2  _  a2  ^  0 ;      •     •     .     •     (336) 

dL  dL  dL 

ax  dy  dz 


MECHANICS    OF    SOLIDS 


229 


and  the  axis  of  z  being  vertical   and 
positive   downwards, 

which    values    in     Equations    (319), 
cive 


dP-x  d?y        . 

y  •  -T-^  —  a;-  -^4-  =  0: 


d'^z 


dt^ 

d~y 
dfi 


1^.(33': 


0 


and  differentiating  the  equation  of  the 
sphere  twice,  we   have 

xd^-x  +  ycf^y  +  2.d^z  =  -  (o?x2  +  dy^  +  d z'^)  ; 

dividing   by  df^,  and  replacing  the   second  member  by  its  value   F-, 
the  velocity,  we  find, 


d~  X  d'^y     , 

X h  y  •  — ~  + 

df"^  ^      de    ^ 


d-^z 


=  -  V. 


But,  Equation  (335), 

V^  =  2gz  +  C (338) 

and  denoting  by  V  and  k,  the  initial  values  of  V  and  z,  respectively, 
we   have 

V2  =  F'2  _(-  2g  {z  -  k), 

which  substituted  above,  gives 

Eliminate  x,  y,  d- x,  d"^  y,  from  this   equation   by   means  of  Equa- 
tions (336)  and  (337). 

From   the  latter  we  find, 


(Py 
df 

d^x 


y    /d^z  \ 

X    /d-  z 


d^x  X    /d'Z  \ 

'dW  ~  T  \dfi         ''''/ 


230  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

which  substituted  in  Equation  (339),  and  reducing  by  means  of 
Equation  (336),  we  get 

a2.^  =g{a^  -Sz^  +  2kz)  -  V'^z; 
a  r 

multiplying  by  2  c?  2-,  and  integrating,  we  find 

a2.  ^  =  2g{a^z  -  z^  +  hz^^)  -  F^  .-2  +  C ; 

in  which  C  is  the  constant  of  integration,  and  to  determine  which, 
we  denote  the  component  of  the  velocity  F',  in  the  direction  of  the 
axis  2,  by  F/,   and   make  z  =  k.     This  being  done,  we  get 


C7  =  a2.F/2  +  F'2F  -  2gu?k; 


whence. 


a2.^=2^(a22-23  +  ^.22)_   7'2  22  4.  ^2  ]^ '2  +   V'^  k'' -  2  g  0?  k, 

adding  and  subtracting  a^  F'2  in  the  second  member,  this   reduces  to 

a2.  ^  z=  (a2  -  g2)  [F'2  -Ig  {k  -  z)]  -\-  C, , 

in  which 

•  C,=  -  {a?  -  k^")  F'2  +  a2  F/2. 

Finding   the  value  of  d  t,  and  integrating,  we  have 

t=f     ,  ""''  .  .  .  •  (340) 

J    ^(^^37^2)  |-p2  -2g{k-  z)]  -h  G, 

Could  this  equation  be  integrated  in  finite  terms,  then  would  z 
become  known  for  a  given  value  of  t ;  and  this  value  of  z  in 
Equation  (336),  and  the  first  of  Equations  (337),  after  integration, 
would  make  known  the  values  of  x  and  y,  and  hence  the  position 
of  the  body ;  its  velocity  would  be  known  from  Equation  (335). 
But   this   integration   is   not   possible. 


MECHAXICS    OF    SOLIDS.  231 

§222. — We  may,   however,    approximate   to   the   result   when   the 
initial    impulse  is   small   and  in  a  horizontal   direction,  and   the  point 
of  departure   is  near  the   bottom   of  the   bowl.     Let  d  be   the   angle 
which   the  .radius  drawn    to  the  variable  position  of  the  body  makes 
with   the  axis   of  z ;    <p,   the   angle   which    the   plane  of  the   angle  6 
makes  with   the   plane   through  the   axis   z,   and   initial   place   of  the.  ^ 
body,  supposed  in   the  plane  xz;    V  =  (3  y/fa,  the  velocity  of  pro- c.^///^,,; 
jection    in    a   horizontal    direction,    /3    being   a  very    small    quantity  ;  *'*'"*'''•'" 
and   a   the    initial    value  of  d.     Then,  because  a  is  very  small, 

k  —  a  cos  a  =  a  (cos^  ^  a  —  sin^  ^  a)  =  a  —  ^  a  a!^  • 

and  for  the  same  reason, 

2  =  a  —  ^  a  .  62  ;     also,     v  =  x  tan  m  : 


V(a2    _   /32)2    _    [-2^2    _    (a2   +   ^2)]2 

whence  by   integration 


2t  - 


r2^2  -  (a2  -f  /32)"| 

L         a2-/ot         J+^>     •     •      (34^) 


making  ;;  =  0,   and   ^  =  a,    we   have    (7  =  cos  M,    whence    (7=0; 
and  solving  the  equation  with  reference  to  ^,  we  get 

^2  =  ^  (a2  +  /32)  +  ^  (a2  _  /32).cos2y^.  <.  .  .  .  (343) 


^30    ELEMENTS  OF  ANALYTICAL  MECHANICS. 

which  substituted  in  Equation  (339),  and  reducing  by  means  of 
Equation  (336),  we  get 

multiplying  by  2dz,  and  integrating,  we  find 

a2 .  ^  =  2^  (a22  -  23  +  ^..2)  _  7'2  ^2  4.  (7 . 
a  r 

in  which  C  is  the  constant  of  integration,  and  to  determine  which, 
we  denote  the  component  of  the  velocity  V,  in  the  direction  of  the 
axis  z,  by  F/,   and   make   z  z=  k.     This  being  done,  we  get 


^-x,^^^^-/--^^.^^-^ 


^ 


^^^   ~_  -  ^^-_i^ _-- r^z —  ^$^ 

Finding   the  value  of  c?  i',  and  integrating,  we  have 

•  •  •  •  (340) 

,/l^f^z^)  [  K'2  -  25r  (A  -  2)]  ^-  C, 

Could  this  equation  be  integrated  in  finite  terms,  then  would  z 
become  known  for  a  given  value  of  t ;  and  this  value  of  z  in 
Equation  (336),  and  the  first  of  Equations  (337),  after  integration, 
would  make  known  the  values  of  x  and  y,  and  hence  the  position 
of  the  body ;  its  velocity  would  be  known  from  Equation  (335). 
But   this    integration   is   not   possible. 


■tj     ei  1 


MECHANICS     OF    SOLIDS.  231 

§222. — We  may,    however,    approximate    to   the   result    when    the 
initial    impulse  is    small   and   in   a  horizontal   direction,  and   the  point 
of  departure   is  near   the   bottom    of  the   bowl.     Let  6  be   the    angle 
which   the  .radius  drawn    to  the  variable  position  of  the  body  makes 
with   the  axis   of  2 ;    9,    the   angle   which    the   plane  of  the   angle  6 
makes  with   the    plane   through  the   axis    z,    and    initial    place    of  tlio.    e 
body,  supposed  in   the  plane  xz;    V  =  f3  ■^fya,  the   velocity  of  pro- ^. >-/.// ^„: 
jection    in    a   horizontal    direction,    /3    being   a  very    small    quantity  ;  *"'"*'''''" 
and   a   the    initial    value  of  L     Then,  because  a  is  very  small, 

k  ^^  a  cos  a  =  a  (cos^  ^  a  —  sin^  ^a.)  =  a  —  ^aa?- 

and  for  the  same  reason, 

2  =  a  —  ^  a  .  fl" ;     also,     y  =  x  tan  <p  ; 

after  neglecting  \oC^  in    comparison  with    unity, 


dt         d  t    dz 

.    '■^i 

T^~Tz'T^^ 

-  —  a-&-  —- 
dz 

and    substituting   the    value    of  the   last  factor  from  Equation  (340), 
dt  _  fa  6 

■which  may  be  put  under   the   form 

V  y   >y    ^/{a?  —  ^2)2  _   [^2  ^2  _  ^a2  ^  ^2)-|2 

whence   by   integration 


o  ,         A         _,  r2  ^2  _  U2  4.  /32)-| 
2t^^-.co.\—-^\^^\+C;     .     .     (34^) 

making  t  =  0,   and   6  —  a,    we   have    C  =  cos~^  1,    whence    (7=0; 
and  solving  the  equation  with  reference  to  6,  we  get 


^2  =  ^  (a2  +  /3=)  -f  ^  (a2  _  /32).cos2>y?.  f. 


(343) 


232  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

From  which  it  appears  that  the  greatest  and  least  vahies  of  6, 
will  occur  periodically,  and  at  equal  intervals  of  time.  The  former 
of  these   values   is   found   by   making 

cos  2  \  /--  -t—l:     whence  2\/-- 1  =  0,     or  =  2  *,     or  ~  4  -r, 
\  a  \  a 

and  so  on;  and  for  a  single  interval  between  two  consecutive  maxi- 
ma, without  respect   to   sign, 

t  =  *x  A; (344) 

^  y  9 

K,7//^   off 

the  maximum  ^being  a. 

The  least   value   occurs   when 

cos  2  v/-  •  <  =  —  1,     or  2\/-^=  rf,     or  =  3  cr,    &c. 
V  a  V  a 

whence  for  a  single  interval  between  any  maximum  and  the  succeed- 
ing minimum, 

i  =  ^<^-,      . (345) 

the   minimum  ^being   ^. 

The  movement  by  which  these  recurring  values  are  brought  about, 
is  called  oscillatory  motion;  that  between  any  two  equal  values  is 
called  an  oscillation;  and  when  the  oscillations  are  performed  in 
equal    times,    they   are    said   to   be   Isochronous. 

Again, 

<f  (p        d(p    d  t  ^ 
d6  ~  dt     dd^ 

substituting  for  -r-,  its  value  obtained  from  the  relation  y  =  a;tan<p, 
we  find 

rfO-"r-a;2+y2'V  dt  -^         dt^       dd 

Integrating  the  first  of  Equations  (337),  we  get 

^     dx        ^     dy        ^       a^„  O-  r,     / — 


M 


ECHANICS     OF    SOLIDS. 


dt 
substituting   this   above,  and  also  the   value   of  -jj ,  giv 

tion  (341),  we  find 


233 
en  by  Equa- 


te 9 
~dJ 


a./3 


a  ^  (a2  _  ^2)  ^^2  _,  ^2j 

dividing  this  by  Equation  (341), 


(346) 


'dt 


'9    «>^ 
a       (3- 


a./3 


/:: 


£  ^Z 


^j -^  ^   s^^^^'i'ldf^^^f  . . 

,; — - 

\cr  ^^^  z^  jii/^ 


^^ 


,' — — 


trom  which   the   azimuth   of  the   plane    of    oscillation   may   be   found 
at   the   end   of  any  time. 

Making   tan  9  =  gd,  we   have 

^.;  =  — •rr;     or     =17*;     or     nz— cr,  &c., 


232  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

From  which  it  appears  that  the  greatest  and  least  values  of  6, 
■will  occur  periodically,  and  at  equal  intervals  of  time.  The  former 
of  these   values   is   found   by   making 

cos2\/--t=l;     whence  2 v /- •  ^  =  0,     or  =  2 *,     or  =  4 *, 
\   a  V  a 

and  so  on;  and  for  a  single  interval  between  two  consecutive  maxi- 
ma, without  respect   to   sign, 

i  =  <<  /^; (344) 


fT^  -  "'"  a;2  +  y2"-     V     dt         ^      dt^     dd 
Integrating  the  first  of  Equations  (337),  we  get 


MECHANICS     OF    SOLIDS.  033 

substituting   this    above,  and  also  the    value    of  -r— ,  given  by  Equa- 

(10 

tion  (341),  we  find 

l^=- "-^  .;      ....    (34G) 

dividing  this  by  Equation  (341), 
d(p  fg    a.jS  fg  a .  (3        


(It         \    a       d-  V   a 


l-  (a2  +  (3^~)  +  ^  (a2  _  ^2)  .  0032^/^  •  « 

but  rz-H^'^-  ^^^•v/?v/c^>9^>-^^-'^/f''.^y- 

^75^  cos2\/— -^  =  cosS.a/—  .?  —  sin^i /— .  ^; 

V  a  V   a  V   a 


whence 


rf(p  fff  « •  /3 


d  t  V   a  fa  ^  „     /  9 

a?  '  cosH  /—  •  <  +  /S2  .  sinH/—  •  ^ 
V   a  V    o 


from  which  we  find 


(347) 


^  -        -^^^ 


a 


d<p  = 


1+^.tan^.  A., 
a''  V    a 


and   integrating,  '■  - 

tan  cp  =  —  .  tan  \  /—  •  ^ (348) 

a.  \    a 

from  which   the   azimuth    of  the   plane    of    oscillation   may   be   found 
at   the   end   of  any  time. 

Making   tan  9  =  oo,  we   have 

71  3  5        . 

'      ^  =  — ^;     or    =-5-*;     or     =  —  ir,  &c., 


23-i  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

and   the   interval  from  the  epoch  to   the   first  azimuth  of  90°,  is 

1  fa 

2  M   g 

and   to   the   first   azimuth   of  270°, 

3  /a" 

and  the  interval  from  the  azimuth  of  90°  to  the  next  azimuth  of  270°, 

[a 

t,,     —    t.     =     t    =    >!(  •  \       5 

V    ^r 

equal    to   the    time  of  one    entire   oscillation. 

From    Equation    (348)    we    have,    after   substituting    for   tan  (p    its 
value   in   the   relation   y  ■=  x  tan  9, 

^2^2    -  t^^   V  a     '' 
adding   unity  to   both   members, 

F^^  -  1  +  tan   y  ^  .  ^ 

also    from   y  =r  .r  .  tan  9, 

3.2      _|_      y2 


^.2 


1  +  tan^  9  ; 


dividing   the  last  equation  by  this   one,  and  replacing  x^  -\-  y^  by  its 
value  a^  —  g^,  from   the   equation   of  the   surface,  we  get 

1  +  tan2  -v/^  •  t 
a?y^  +  /3^^-^  =  ^^-(a^  -  ^^) '  I^  tan'.  9     " ' 

but,  neglecting  the  term  involving  ^*, 

a2  _  s2  _  ^2  ^2  . 

substituting   this    above,  replacing    tan2  9    by    its    value    in    Equation 
(348),  and  ^2  by  its  value   in  Equation  (343),  after  making 

cos  2  V  /-=—  •  /  =  cos^  \     —  •  t  —  sin^  \  / ■^—  •  t. 


MECHANICS     OF     SOLIDS, 
and   rcdnclMg   Ly   the   relation, 


235 


COS'' 


t  +  sin^ 


t  =  1 


we    nave 


ha 


cc2    ^   /:^2         «  ' 


(349) 


•which  shows  that  the  projection  of  the  path  of  the  body  on  the 
plane  a;?/,  is  an  ellipse  whose  centre  is  in  the  vertical  radius  of  the 
sphere,  and  that  the  line,  connecting  the  body  with  the  centre  of 
the   sphere,  describes    a   conical    surface. 

If  a  =  /3,  then  will,  Equations  (343)  and  (348), 


^2 


„2    _    ^2.        (J,    _   W-^.  t; 


^ 


^e?-    J^.-i 


/^-^3  ^^v^.^ 


■^^  zJli 


'7-$^'' 


^' 


u^-o.^^/A^^i-fi^oBL^^t^^r 


z. 


.     .     (350) 
a    uniform 

body's  path, 
s  centre  of 


— :  c-c^ 


from  the  bottom  point  A 
of  the  bowl,  by  a  force  which 
varies  inversely  as  the  square 
of  the  distance  ;  required  the 
position  of  the  body  in  which 
it  would   remain  at   rest. 

As  the  body  is  to  be  at 
rest,  there  will  be  no  inertia 
exerted,  and  we  have 


=  0 


dfi 


=  0 


=  0 


234:  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

and   the   interval  from  the  epoch  to   the   first  azimuth  of  90°,  is 

1  [^ 

and   to   the   first   azimuth   of  270°, 

3  /a" 

and  the  interval  from  the  azimuth  of  90°  to  the  next  azimuth  of  270°, 

a 

— 1 
9 

equal   to   the   time  of  one   entire   oscillation. 

From    Equation    (348)    we    have,    after   substituting    for    tan  (p    its 
value   in   thf'   i-plntinn    ?y  r:=  X  tan  (D. 


adding   unit 


also    from 


dividing   th. 

value  <J?  —  s;2,  from    the    equation    of  the    surface,  we  get 

1  +  tan2  sj  —  •  t 

o?,f  +  /32.f2  =  /32.(a2  -  ^2)  _^ ^ . 

1  +  tan2  9 

but,  neglecting  the  term  involving  ^*, 

a2  —  22  =  a2  ^2 . 

substituting   this    above,  replacing    tan2(p    by    its    value    in    Equation 
(348),  and  ^2  \^^  its  value   in  Equation  (343),  after  making 

cos2\/-^  •  t  =  cos2  \     —  •  t  —  sin2  \/ -^—  •  t, 
V    a  V    a  V    a 


MECHANICS     OF     SOLIDS, 
and   reducing   by    the    relation, 


235 


we    have 


■^  '  t  -\-  sin^ 
a 


•  ^  =  1 


(349) 


■which  shows  that  the  projection  of  the  path  of  the  body  on  tlio 
plane  xy,  is  an  ellipse  whose  centre  is  in  the  vertical  radius  of  the 
sphere,  and  that  the  line,  connecting  the  body  with  the  centre  of 
the    sphere,  describes   a   conical    surflice. 

If  a  =:  /3,  then  will,  Equations  (343)  and  (348), 


^2  -  a2  =  /32  ; 

and,  Equation   (349), 

a;2  +  2/2  =  a2  a2 ; (350) 

hence,    the    body   will    describe   a    horizontal    circle   with    a    uniform 
motion. 

The   pressure  upon   the    surface,  at  any  point  of  the   body's  path, 
is   given  by  the  value  of  iV  in  Equation  (334). 

§223. — Example  2. — Let  the  body,  still  reduced  to  its  centre  of 
inertia  and  acted  upon  by  its 
own  weight,  be  also  repelled 
frona  the  bottom  point  A 
of  the  bowl,  by  a  force  which 
varies  inversely  as  the  square 
of  the  distance  ;  required  the 
position  of  the  body  in  which 
it  would   remain  at   rest. 

As  the  body  is  to  be  at 
rest,  there  will  be  no  inertia 
exerted,  and  we  have 


1^ 


=  0 


dfi 


=  0 


rf2£ 

772 


=  0 


236  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and  assuming  the  axis  z  vertical,  positive  upwards,  and  the  origin 
at   the  lowest  point  A^ 

Z  =  a;2  +  y/2  +  ^2  _  2^2  =  0,     .     .     .     .     (351) 

dx  dy  ^'      dz  ^  " 

and  denoting  the  distance  of  the  body  from  the  lowest  point  by  r, 
the  intensity  of  the  repelling  force  at  the  unit's  distance  by  F,  and 
the   force   at   any    distance  by  P,  then  will 

P=-^;     r  =   ^x"  -\-y'  +  z-^\  •     •     •     •     (352) 

X  r>       y  2         -        , 

for   the  force  P,    cos  a  =   — ;     cos  /iJ  =  — ;     cos  7  =  —  ;     for     the 

weight  My,  cos  a'  =  0 ;     cos  /3'  =  0  ;     cos  7'  =  —  1  ;    and 

Fx  Fv  Fz 

These   several  values  being  substituted  in  Equations  (319),  give 
Fyx        Fy% 


r3 


=  0, 


The  first  equation  establishes  no  relation  between  x  and  3/,  since 
the  equilibrium  which  depends  upon  the  distance  of  the  particle 
from  the  source  of  repulsion,  would  obviously  exist  at  any  point 
of  a  horizontal  circle  whose  circumference  is  at  the  proper  height 
fi'om   the  bottom. 

From   the   second   equation   we  deduce, 

4-  =  -^ (^^^) 

M  g        a 


MECHANICS     OF    SOLIDS.  237 

from  which  r  becomes  known ;  and  to  determine  the  position  of  the 
circle  upon  which  the  body  must  be  placed,  we  have,  by  making 
X  =::  0  in  Equations  (351)  and  (352), 

■y/z^  +  y2  _  ^^ 

2/2  +  s2  _  2  a  2  =  0. 

Equation  (353)  makes  known  the  relation  between  the  weight 
of  the  body  and  the  repulsive  force  at  the  unit's  distance ;  the  in- 
tensity of  the  force  at  any  other  distance  may  therefore  be  deter- 
mined. 

If  there  be  substituted  a  repulsive  force  of  different  intensity, 
but  whose  law  of  variation  is  the  same,  we  should  have,  iu  like 
manner, 

F'     _  r'^ 
Mg    ~  '7' 

hence, 

F :  F  :  :  r^  :  r'^ ; 

that  is,  the  forces  are  as  the  cubes  of  the  distances  at  which  the 
body   is   brought   to   rest. 

If,  instead  of  being  supported  on  the  surfoce  of  a  sphere,  the 
body  had  been  connected  by  a  perfectly  light  and  inflexible  line 
with  the  centre  of  the  sphere  and  the  surface  removed,  the  result 
would  have  been  the  same.  In  this  form  of  the  proposition,  we 
have   the  common  Electroscope. 

The  differential  co-efficients  of  the  second  order,  or  the  terms  which 
measure  the  force  of  inertia,  being  equal  to  zero,  Equations  (33*2), 
show  that  the  rcsiiltant  of  the  extraneous  forces,  in  this  case  the 
weight  and  repulsion,  is  normal  to  the  surfoce,  which  should  be  the 
case ;  for  then  there  is  no  reason  Avhy  the  body  should  move  in 
one  direction  rather  than  another.  The  pressure  upon  the  surface  is 
given    by    the    value   of  N,  in  Equation   (334). 

g  224. — Exanrple  3.     Let   it   be    I'cquired  to  find  the  circumstances 


238 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


of  motion   of  a  body   acted   upon   by   its   own  weight  while   on   the 
arc  of  a  cycloid,  of  which 
the  plane   is   vertical,  and 
directrix  horizontal. 

Taking  the  axis  of  z, 
vertical ;  the  plane  z  a;,  in 
the  plane  of  the  curve ; 
and  the  origin  at  the  low- 
est  point,  then   will 


=  ^  — •v/2a2  ~  z^  —  a  versin    ^ —  ; 


in   which  z   is   taken  positive   upwards. 

dL 


dL 
d  X 


1; 


dz 


v- 


X=0;     Z=  -Mg, 
and  Equation  (330)  becomes 


(354) 


(355) 


d"^  X         /2a 


—  z  d."^  z 


(356) 


From   the  equation   of  the  curve  we   find 


dx  ■=.  d . 


'la   —  z 


whence 


c?x2  4-  dz"^    _  dz"-    2a  _ 

dl^  ~  Je'T  ^   ^^' 


But,  Equation  (335), 


F2:=  -2gz+  C; 


(357) 


(358) 


and  supposing   the   body    to    start   from  i/,  corresponding    to    which 
z  =z  h;  we  have 


0  =  -2gh+  C, 


MECHANICS    OF    SOLIDS. 


239 


and  by  subtraction, 


•vvnence, 


\ 


d  z^     a 


(359) 


Differentiating  Equation  (857),  and   dividing   by   d  t"^,   we  have 
2  a  —  z  a 


d^  X        d^  z 


/ 


2a  —  z 


d_z^ 
dC- 


x/^ 


z 


1C   -  !»C< 


^^' '^^Tf  ^ -JJ^  V  ^  AV^^  ;^/-^- ~^y 


—Iffc^  -  ^^--5.  ^^-^   /2-<«  -^ 


in   the  vcrti- 


Findmg  the  value  of  (fi',  taking  the  negative  of  the  double  sign, 
because  2  is  a  decreasing  function  of  the  time  t,  and  integrating,  we 
have 

dz  I 

"^  ~  ""V  g 


=  -  v/-  •/ 


9      ^      -y/hz   —   Z"" 

Making  z  =  h,  we  have 


a  .  -\  2z       ^ 

vcrsin     •— -  +  C. 
h 


0  =  —  \  /—  •  versin    ^  2  +  C ; 
'J 


238 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


of  motion   of  a  body   acted  upon   by   its   own   weight  while   on   the 
arc  of  a  cycloid,  of  which 
the  plane   is   vertical,  and 
directrix  horizontal. 

Taking  the  axis  of  z, 
vertical ;  the  plane  z  x,  in 
the  plane  of  the  curve; 
and  the  origin  at  the  low- 
est  point,  then   will 


=  X  —sjlaz  —  z^  —  a  versm     —  ; 


(354) 


in   which   z  is 


and  Equation 


From   the  ( 


whence 


i<«^. 


z3&-.. 


^~-~ 

~^^ 


/z^.'z-z* 


dx^  +  dz^   _  dz^    2a  _ 


But,  Equation  (335), 


V'=  -2gz-\-  C', 


(358) 


and  supposing    the   body    to    start   from   Jf,  corresponding    to    which 
z  ^  h;  we  have 

0  =  -2gh+  C, 


MECHANICS    OF    SOLIDS.  239 

and  by  subtraction, 

dz"^     2a 

■vvnence, 

^■f  =  ^("-^'- <3^»> 

Differentiating  Equation  (357),  and   dividing  by   dt"^,   ^ve  have 


''X        d-'z 

2a  —  z 

z                   dz^ 

a 

l^          d(^ 

/2a  -z         ^^' 

/2a-  z 

Avhich  in  Equation  (356),  gives 

d^z  2a  -  z  _^_^,  <^g  _  Q. 

eliminating  -—  ,   by   Equation  (359),  and  reducing, 

dt^-2a''"'       ~'^^' 
multiplying  by  2dz,  and  integrating, 

in  which  (7  is  zero,  because,  when   z  =  A,  the   velocity   in   the  verti- 
cal  direction  will   be  zero. 

Finding   the  value  of  d  t,  taking   the  negative  of  the   double  sign, 

because  z  is  a  decreasing  function  of  the   time  t,  and  integrating,  we 

have 

fa       r         dz  FaT  .  -\2z        ^ 

t  =  —  \  /  —  •  /  — —  =z  —  \  / vcrsm     •—. — h  C. 

V  y    ^   ^hz-z-^  ^9  ^ 

Making  z  =  A,  we  have 

fa  .  -  ^  r,    ,    n 

Q  —  —\    —  ■  versin      2  +  C ; 
V  9 


240  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

whence, 

la 

V  g 

and 


t  =  \    — (*  — versin     •-— ) (3G0) 

y   g    \  h/ 

When   the   body  has  reached  the  bottom,  then  will  z  ==  0,  and 

a 
9 

which  is  wholly  independent  of  A,  or  the  point  of  departure,  and 
we  hence  infer  that  the  time  of  descent  to  the  lowest  point  will  be 
the  same  in  the  same  cycloid,  no  matter  from  what  point  the  body 
starts. 

Whenever  z  =  h,  the  body  will.  Equation  (359),  stop,  and  we 
shall   have   the  times   arranged  in  order  before   and   after  the   epoch, 

—  4ff\/— :     —  S'T*/ — :    0:     2*\/ — ;     4';r\/^5    &c., 
V^r'  V^''  \  g  \  g 

the   difference   between  any  two   consecutive  values   being 

fa 
2if\/  — 
V   g 

The  body  will,  therefore,  oscillate  back  and  forth,  in  equal  times. 
The   cycloid  is,  on   this   account,  called  a  Tauiochronous  curve. 

The   pressure    upon    the    curve   is   given   by  Equation  (334). 

The  time  being  given  and  substituted  in  Equation  (360),  the  value 
of  z  becomes  known,  and  this,  in  Equations  (358)  and  (357),  will 
give   the   body's  velocity  and   place. 

g225. — Example  4. — Let  a  body  reduced  to  its  centre  of  inertia, 
and  whose  weight  is  denoted  by  W,  be  supported  by  the  action 
of  a  constant  force  upon  the  branch  E H  oi  an  hyperbola,  of  which 
the  transverse  axis  is  vertical,  the  force  being  directed  to  the  centre 
of  the   curve.     Required   the   position  of  equilibrium. 


MECUANICS     OF     SOLIDS. 


241 


Denote  the  constant  force  by  W,  which  may  be  a  weight  at  the 
end  of  a  cord '  passing  over  a  small  wheel 
at  C,  and  attached  to  the  body  M.  De- 
note the  distance  CM  by  r,  and  the  axes 
of  the  curve  by  A  and  B.  Take  the  axis 
z  vertical,  and  the  curve  in  the  plane  zz. 
Make 

F'  =  W, 

P"  -  W 
then  will 


cos  7'  =  1,     cos  a'  =  0, 

II              ^               It              "^ 
cosy     = 5     cos  a'  = J 


X=  F'  cos  a'  +  F"  cos  a"  =  -  W'.—, 


Z  =  F'  cos  7'  +  F"  cos  y"  =  W  -  W '  — , 

r 

and   as   the   question    relates  to  the   state   of  rest, 
d^  X        ^        (P  z 

JF  ■--''■'   IF-"- 

The   Equation   of  the   curve   is 

L  =  A^x^  -B^z-^  +  A^  B^  =  0; 
whence, 

dL 


d  X 


=  2A^x, 


dL 
dtz 


-2^2,. 


these  values  substituted   in  Equation  (330),  give 


whence, 


WB^  —  -  WA^x  +  TF.P  —  =  0- 
r  r 


(.12  -{-  B^-)  W'Z  -  WA~  r  =  0 
16 


(301) 


24:2  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

But 

r2   =   X2   +    22    =   S2   +    _,2    _   ^2^32 J^ J^2. 

whence  denoting  the  eccentricity  by  e, 


r  =  ^e'  z2  _  £2 
and   this,   in  Equation   (361),  gives  after  reduction,     | 

B  .  W 


e(lF2_ir'2e2j2 

which,  with  the  equation  of  the  curve,  will  give  the  position  of 
equilibrium. 

If  W  e  be  greater  than  TF,  the  equilibrium  will  be  impossible- 
If  W  e  =  W,  the   body  will  be  supported  upon  the  asymptote. 

The  pressure  upon  the  curve  is  given  by  Equation  (334). 

§  226. — Example  5. — Required  the  circumstances  of  motion  of  a 
body  moving  from  rest  under  the  action  of  its  own  weight  upon  an 
inclined   right   line. 

Take  the  axis  of  z  vertical, 
the  plane  z  x  io  contain  the 
line,  and  the  origin  at  the 
point  of  departure,  and  let  z 
be  reckoned  positive  down- 
wards.    Then  will 


L  :=z  z  —  ax  =z  0, 


d  L 


-  1 


dL 
d  X 


—  —  a\ 


X  =  0  ;     Z  -  Mg; 
which   in  Equation    (330)  give,  after  omitting  the    common  factor  M^ 


d"^  X  d"Z^ 

A-  ao  —  a r=  0.    • 

df   ^  ^  dt^ 


(362) 


From  the  equation  of  the  line  we  have 

d-z 
d"^  X  =.  ; 


MECHANICS     OF    SOLIDS.  243 


which  in  Efjuation  (3G2),  after  slight  reduction,    t 


0?  z 


dfi    ~   1  +  a2   ^ 
Multiplying  by  2dz,  and   integrating, 

dz"^         ^         «- 

the   constant   of  integration   being   zero. 
Whence 

..-./ 'y^ 


/2(l  +  g^ 

~  V        9  •  a2 


9-a^  2  -/  • 

and 


'^  >;    and    if  we 

2.  =  ^:r1j^ . 


'/p^ 


e'^l^^ 


in    which  d  denotes  the  distance  A  C. 

But  the  second  member  is  the  time  of  falling  freely  through  the 
vertical  distance  d;  if,  therefore,  a  circle  be  described  upon  A  C  as 
a  diameter,  we  see  that  the  time  down  any  one  of  its  chords,  ter- 
minating at  the  upper  or  lower  point  of  this  diameter,  will  be  the 
same  as  that  through  the  vertical  diameter  itself.  This  is  called  the 
mechanical   property  of  the  circle. 

Example  0. — A  spherical  body  placed  on  a  plane  inclined  to  the 
horizon,  would,  in  the  absence  of  friction,  slide  under  the  action  of 
its  o^vn  weight;  but,  owing  to  friction,  it  will  roll.  Required  the 
circumstances  of  the    motion. 


242  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

But 


whence  denoting  the  eccentricity  by  e, 


and   this,    in  Equation    (361),  gives  after  reduction,     | 
_  B  .  W  . 

which,    with  the    equation    of    the   curve,    will   give   the    position   of 
equilibrium. 

If    We  be  greater  than    W,   the   equilibrium   will   be   impossible- 
If  We  =  W,  the   body  will  be  supported  upon  the  asymptote. 

The    preSSU^''^    \iT\r\n     +Vio    nMr-^ra    ^a    n-.'Tron     \\\r    TTnnQfmn     /S5i4.> 

§  226.—^^       _  ^  ^2 
inclined   right  '"^^ 


body  moving  '^  .^^^^ 


^;.<f^^ 


/i^^y^^y 


/t'/* 


J5 


VS 


/, 


Take  the  f 
the  plane  z  x 
line,  and  th( 
point  of   depj 

be     reckoned    positive    down- 
wards.    Then  will 

L  ^^  z  —  ax  =  0, 

d  L         ,,    dL   __ 

dz  '     dx  „\ 

X=0;     Z  =  Mg; 
which   in  Equation    (330)  give,  after  omitting  the   common  factor  M, 

...      (362) 


-i- 


d^x 


d"Z 


4-  ag  —  a  ——r  —  0. 
dt^  ^  dfi 

From  the  equation  of  the  line  we  have 

d"^  X  =1  — —  ; 


MECHANICS     OF    SOLIDS.  243 

which  in  Equation  (3G2),  after  slight  reduction,    t 


dfi 

a2 

~   1  + 

a? 

'9 

Multiply 

ing 

by 

2dz, 

and   iutc 

dz-^ 
dfi 

grating, 

Cl^ 

-"^l 

■f 

d" 

the   constant   of  integration   being   zero. 
Wh'ince 

dt--' ^      '' 


/2  (1  +  «^ 
~  V         q-d"- 


2V. 


and 


\         (JO?  \       g  a^z 


the  constant   of  integration  being  again  zero. 

The   body  beuig   supposed   at  B^  then  will   z  =  AD\    and    if  we 
draw  from  B  the  perpendicular  B  C  to  A  B,  we  have 

2  \  V  V  s».>.  V  2 

AB     _  1  +  a 
which  substituted  above. 


AB     2  /2rf 


(364) 


in    which  (/  denotes  the  distance  A  C. 

But  the  second  member  is  the  time  of  falling  freely  through  the 
vertical  distance  rf;  if,  therefore,  a  circle  be  described  upon  -4  C  as 
a  diameter,  we  see  that  the  time  down  any  one  of  its  chords,  ter- 
minating at  the  upper  or  lower  point  of  this  diameter,  will  be  the 
same  as  that  through  the  vertical  diameter  itself.  This  is  called  the 
mechanical   property  of  the  circle. 

Example  6. — A  spherical  body  placed  on  a  plane  inclined  to  the 
horizon,  would,  in  the  aljsence  of  friction,  slide  under  the  action  of 
its  own  weight;  but,  owing  to  friction,  it  will  roll.  Required  the 
circumstances  of  the   motion. 


2M 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


If  the  sphere  move  from  rest  with  no  initial  impulse,  the  centre 
will  describe  a  straight  line 
parallel  to  the  element  of 
steepest  descent.  Take  the 
plane  0:2,  to  contain  this 
element,  the  axis  z  vertical 
and  positive   upwards. 

The  equation  of  the  path 
will   be. 


whence, 


L  ■=.  z  -\-  X  tan  a  —  A  =  0 ; 


dL        ,        dL       ^ 

—r—  =  1  ;     —r~  =  tan  a. 
dz  ax 


The  extraneous  forces  are  the  weight  of  the  sphere  and  the  fric- 
tion. Denote  the  first  by  W,  and  the  second  by  F.  The  nature 
of  friction  and  its  mode  of  action  will  be  explained  in  the  proper 
place,  §  307 ;  it  will  be  sufficient  here  to  say  that  for  the  same 
weight  of  the  sphere  and  inclination  of  the  plane,  it  will  be  a  con- 
stant force  acting  up  the  plane  and  opposed  to  the  motion.  We 
shall   therefore   have 

Z  =z  —  Mg  -{■  Fsin  a  ;     X  =  —  Feos  a, 

which   values,  and   those   above   substituted   in   Equation  (330),  give 

-  Fcosa  -M-^  +  (Mg^+  ^~)  tana  =  0. 


But  from   the   equation  of  the   path,  we   have 
d'^z  =.  —  d'^  X-  tan  a  ; 
and    eliminating   d'^x  by  means  of  this   relation,  there   will   result 
d^z         .        /F       ^..  .      \ 


MECHANICS    OF    SOLIDS.  245 

Multiplying   by   2  c? 2,    integrating    and   making   the   velocity    zero 
when  z  =  h,  vre   have 

-—  =z  F2  =  2  sm  a  (  —  \  oo^«?  —  g  sm  a)  •  (2  —  h). 

This  gives 

1  dz 

dt  = 


\/2  sin  a  (^—  veX?tJ^^  —  ^  sin  a  J      "^ 


and   by  integration,  the   time  being  zero  when  z  =.  h, 

F 

h  —  z  =  ^  sin  a  (^  •  sin  a —  •  ^^li^)  •  t"^. 

Again,  all  axes  in  the  sphere  through  its  centre,  are  principal 
axes ;  the  sphere  will  only  rotate  about  the  movable  axis  y,  in 
which  case  v^  and  v,  will  each  be  zero,  and  Eouations  (^228'i  will  give 

Whence, 


Multiplying   by   2  6^%^,  integrating,  and    making   the   angular  velocity 
and  the   arc  4/  vanish   together, 

d^^_  2Fr^ 

df   ~  Mkf^'' 

whence, 

d^ 


[Mkf  d\. 


2M 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


If  the  sphere  move  from  rest  with  no  initial  impulse,  the  centre 
will  describe  a  straight  line 
parallel  to  the  element  of 
steepest  descent.  Take  the 
plane  xz,  to  contain  this 
element,  the  axis  z  vertical 
and  positive   upwards. 

The  equation  of  the  path 
will   be, 


L  =z  z  -\-  X  tan  a  —  A  =  0 ; 


whence. 


dL  _ 
dz    ~     ' 


dL 

dx 


=  tan  a. 


t  - 


The  extraneous  forces   are   the   weight  of  the   sphere  and  the  fric- 
tion,     Y     „ ,  ,  '  '  ,       --      ™ 
of  frictii                                    ' 
place,   § 
weight  ( 

stant   fo  \  <^^ 

shall   th 


y.  ^  ^  »^ 


which   values,  and   those   above   substituted   in   Equation  (330),  give 

d'^  X         /  d"^  z\ 

—  Fcosa  —  M-  -rr  +  ( -^^9  +  ^'  -rr )  tan  a  =  0. 
dfi         \     *'a  d  fi  / 


But  from   the   equation  of  the  path,  we   have 
d'^z  =^  —  d^  X-  tan  a  ; 
and    eliminating   d'^x  by  means  of  this   relation,  there   will   result 


d^s         .       {F        ^^  .      \ 

—  =  sma  \-^xis^^  -  g  sm  a) 


MECHANICS    OF    SOLIDS.  245 

Multiplying   by   2dz,    integrating    and   making   the   velocity    zero 
when  z  =  h,  we   have 

— —  =  F2  =  2  sin  a  \--  \  OO^ifj  —  g  sin  a  J  •  (2  —  K). 

This  gives 

1  dz 

dt  = 


yjl  sin  a  (^—  \eX^^  —  g  sin  a^ 


■\/  z  —  h 


and   by  integration,  the   time  being  zero  when  z  z=  h, 

F 

h  —  z  =  l'  sin  a  (^  •  sin  a —•  ^9"^"*^)  •  t^. 

Again,  all  axes  in  the  sphere  through  its  centre,  are  principal 
axes ;  the  sphere  will  only  rotate  about  the  movable  axis  y,  in 
which  case  v,  and  v,  will  each  be  zero,  and  Equations  (228)  will  give 

dt  " 


wherein, 


dv  (^24. 

'  '      dt  dt^  '         ' 


r  being   the   radius   of  the   sphere. 
Whence, 


fZH  ^r 


df    ~  Mk;^ 

Multiplying   by   2(^4',  integrating,  and    making   the   angular  velocity 
and  the   arc  4/  vanish   together, 

rf42'_  2i^r 
dt^   ~  JTk} 

whence, 


4; 


V  27V 


246  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and  by   integration,  making  t  and  ^^  vanish  together, 

F.r 

Also,    because   the    length  of  path  described  in  the   direction   of  the 
plane  is  r.-]^,  we  have,  in  addition, 

A  —  2  =  r  .  -^z .  sin  a ; 

and    eliminating  4^  fro'Ti    this    and    the    above    equation,   there    will 
result 

t   =\/   ..       2        •  (^    -   ^)- 

If  the   sphere  be  of  homogeneous   density    throughout,    then   will. 
Example  5,  §  182, 

and 


_      /4    M  h  -^ 
"~  V  5     ^    sin  a 


If  the  entire  mass  of  the  sphere  were  concentrated  into  its  surface, 
then  would 

A:/  =  r2, 

and 


N^ 


M    h  -  z 


F      sin  a 
which  values  for  the  times  are  to  each  other  as  -/O,  8   to    -y/^. 

CONSTKAESTED   MOTION  ABOUT   A   FIXED   POINT. 

I  227.— If  a  body  be  retained  by  a  fixed  point,  the  fixed  and 
what  has  been  thus  far  regarded  as  a  movable  origin  may  both  be 
taken  at  this  point;  in  which  cas#,  5x^,  8y^,  Sz^,  in  Equation  (40), 
will  be  zero,  the  first  three   terms  of  that  general    equation    of  equi- 


MECHANICS    OF    SOLIDS.  247 

librium  will  reduce  to  zero  independently  of  the  forces,  and  the  equi- 
librium Aviil  be  satisfied  by    simply  making 

X     (*    7J  —  V  (L^  X 

2  P  (x  cos  /3  —  y  cos  a)  —  2  m  ■  — ~— ^^ —  0 

I.  F{z  cos  a  —  X  cos  7)  —  2  m — =  0  ;   ^  ■  •  ('5^>^) 

^,                   y  .  d^  z  —  z  .  d~  y 
2  P  (y  cos  7  —  ^  cos  /3)  —  2  7?i  • — ■-  ^  0  ; 

the    accents   being    omitted    because    the    elements    ?«,  ?«',  drc,    being 
referred   to    the   same   origin,  x',  y',  z'  will  become  ar,  y,  z. 

The  motion  of  the  body  about  the  fixed  point  might  be  discussed 
both  for  the  cases  of  incessant  and  of  impulsive  forces,  but  the  discus- 
sion being  in  all  respects  similar  to  that  relating  to  the  motion  about 
the  centre  of  inertia,  §  269  and  §  187,  we  pass  to 

CONSTRAINED   MOTION   ABOUT   A   FIXED   AXIS. 

§  228. — If  the   body   be    constrained   to   turn    about   a  fixed    axis, 

both   origins    may    be   taken    upon,    and    the    co-ordinate    axis    y    to 

coincide  with  this  axis;   in  which  case    8x^,     Sy^,    5z^,  Sep    and    S-a, 

in    Equation    (40),    will    be    zero,    and    to    satisfy    the    conditions   of 

equilibrium,    it    will    only    be    necessary    for    the  forces   to    fulfil    the 

condition, 

z  d  X  ~~"  X  •  u^  z 
2  P  (2  cos  a  —  a;  COS7)  —  Im — — ■ —  =  0    •  •  (3G6) 

the    accents  being    omitted  for  reasons  just  stated. 

§220. — The  only  possible    motion   being   that   of   rotation,    lot    us 
transform   the   above    equation    so   as  to  contain  angular  co-ordinates. 
For  this   purpose  we   have.  Equations  (36), 

a;'  =  r"sin-^;     s'  =  r"cos4. (367) 

in  which  r"  denotes  the   distance   of  the  element  m  from  the  axis  y. 
Omitting  the  accents,  differentiating  and  dividing   by  d  t,  we  have 

—  =  r  cos  4'  c^4'  j     -5-  =  —  r  sin-l'  •  f^4'  *     *     '     (368) 
d  t  d  i 


248  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Now, 

z-d'^x        X'cPz         1      ,/      dx  dz^ 


x-d^-z         1      .  {     dx  dz\ 

df'  dt        \      dt  dt/  ' 


di^ 

whence  by  substitution,  Equations  (367)  and  (368), 


d'^x  d^z  1      ,  /  2    ^4'\         2    ^^  + 


d¥~  '"''difi  -  dt    y  '  dtJ  -     di^ ' 


and   since    — -    must  be  the  same  for  every  element,  we  have,  Equa- 
tion   (366), 

I.m  r^ '  —r-^  =  IP  {zQOsa  —  X  cos  7), 

and 

^li  —  ^  -P  •  (g  cos  g  —  ar  cos  7)      ,     ,     .     ,     /g^gx 
df^    ~  -Zmr"^  .     .     .     .     ^       ; 

That  is  to  say,  the  angular  acceleration  of  a  body  retained  by  a 
fixed  axis,  and  acted  upon  by  incessant  forces,  is  equal  to  the 
moment  of  the  impressed  forces  divided  by  the  moment  of  inertia 
with   reference   to   this  axis. 

Denoting  the  angular  velocity  by  F, ,  and  the  moment  of  inertia 
by  7,  we  find,  by  multiplying  Equation  (369)  by  2d-\^  and  integrating, 

IVi"^  =  2f'S.P{zcosa  —  xQosy)d-\.  -{-  C,     " 

and   supposing   the   initial  angular   velocity  to  be   F/,  we  have 

/(F,2  —  F/2)  =  2  f^P  {zcosa  -  xcosy)d-].. 

But  the  second  member  is,  §  105,  twice  the  quantity  of  work 
about  the  fixed  axis ;  whence  the  quantity  of  work  performed  be- 
tween the  tM'O  instants  at  which  the  body  has  any  two  angular 
velocities,  is  equal  to  half  the  difference  of  the  squares  of  these 
velocities  into  the  moment  of  inertia,  or  to  half  the  living  force 
gained    or   lost    in    the   interval. 


MECHANICS    OF    SOLIDS. 


249 


If  Fj-  —  F/-  =  1,  we  find  the  value  of  /  to  be  twice  the 
quantity  of  work  required  to  produce  a  change  in  the  square  of  the 
angular  velocity  equal   to   unity. 


COMPOUND    PENDULUM. 

§  230. — Any   body   suspended  from   a   horizontal   axis  A  B,    about 
which   it   may    swing  with  freedom    under    the 
action  of  its  own  weight,  is  called  a  compound 
pendulum. 

The  elements  of  the  pendulum  being  acted 
upon    only  by  their  own  weights,  we  have 

P  =  mg  ;     P'  =  m'^,  (i:c.  ; 

the  axis  of  z  being  taken  vertical  and  positive 
downwards, 

cos  a  =r  cos  a'  =  &c.  =  0  ; 

cos  V  =:  cosy'  =z  &C.  =  \, 

and  Equation  (369)  becomes 

2  mar 


~~dfi 


9 


2  m  r"^ 


(370) 


Denote  by  e,  the  distance  A  G,  of  the  centre  of  gravity  from  the 
axis;    by  nJ.*,    the   angle  HAG,  which 
A  G  make*  with  the  plane  yz;    by  ;r^, 
the    distance  of  the   centre  of  gravity 
from  this  plane ;    then  will 

x^  =^  e  .  sin  %)./  ; 

and  from  the   principles  of  the  centre 
of  gravity, 

2  m  .r  =  Mx^  =  M .  e  .  sin  4- ; 
which  substituted  above,  gives 


d^ 


=  —  9 


M.  e .  sin  4^ 


2  m  )•'■ 


(371) 


250  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Multiplying  by  2d-^,  and  integrating, 

-T72  =  ^9--^ ^-0084.  +  G. 

a  t^  2  m  r^ 

Denoting  the  initial  value  of  4-  by  a,  we  have 

0  ^  ^9'^ 5 -cos  a  +  (7; 


whence, 


but 


cos4.  =  l-j±^+__fel__&c. 


cos  a  =:  1  —  :: — -^  +  - — ^^—^ — r  —  &C. 


and  taking  the  value  of  •>]>,  so    small   that  its   fourth   power   may  be 
neglected  in  comparison  with  radius,  we  have 

COS  4^  —  COS  a  = ; 

which  substituted  above,  gives,  after  a  slight  reduction,  and  replacing 
2  711  r^  by  its  value  given  in  Equation  (244), 

J4. 


V       e.a 


dt 


■ 
the  negative  sign  being  taken    because  4^  is  a    decreasing  function   of 

the  time. 

Integrating,  we  have 

/^,2  4-  e2  14. 

t=z\/- COS        — \^'^) 

\       e.g  a 

The   constant   of   integration  is   zero,  because  when  \  =.  a.^  we    have 
i  =  0. 


MECHANICS    OF    SOLIDS.  251 

ijj  y  :=z  —  a,  we  have 


'\/S 


i^; (374) 


Avhich  gives  the  time  of  one  entii-e  oscillation,  and  from  which  we 
conclude  that  the  oscillations  of  the  same  pendulum  will  be  isochro- 
nal, no  matter  what, the  lengths  of  the  arcs  of  vibration,  provided 
they  be   small. 

If  the  number  of  oscillations  performed  in  a  given  interval,  say 
ten  or  twenty  minutes,  be  counted,^ the  duration  of  a  single  oscillation 
will  be  found  by  dividing  the  whole  interval  by  this  number. 

Thus,  let  ^  denote  the  time  of  observation,  and  N  the  number  of 
oscillations,  then  will 


^..  /EjE 


V^V" 


other  location 

different,  the 

hall   have,   as 


\/7Z 


iV2    -   g 


(374)' 


that  is  to  say,  the  intensities  of  the  force  of  gravity,  at  different 
places,  are  to  each  other  as  the  squares  of  the  number  of  oscilla- 
tions performed  in  the  same  time,  by  the  same  pendulum.  Hence, 
if  the  intensity  of  gravity  at  one  station  be  known,  it  will  be  easy 
to  find  it  at  others. 

§  231. — From  Equation    (372),  we  have 

-—  •  2  m  r-  =  2  M .  g  .  e  (cos  4^  —  cos  a)  ;     .     •     (375) 


250  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Multiplying  by  2d-^,  and  integrating, 

fZ4.2  M.e.  ,     ,     ^ 

-T7i  =  ^ff--^ ^•cos4.  +   C. 

dp  2  m  r^ 

Denoting  the  initial  value  of  ■\'  by  a,  we  have 

Me  ,     „ 

0  =  2  a  • •  cos  a  +  (J; 

2  m  r^ 


whence, 


but 


d-l"^       ^       M.e 


4.2  14 


&c. 


and  taking  t 
neglected  in 


which  substiti 
2  m  r^  by  its 


-/ 


at-     ■/"'+'' 


the  negative  sign  being  taken    because  4^  is  a    decreasing  function  of 
the  time. 

Integrating,  we  have 

t  =  \/~^ cos      — {■^'■^) 

V       e.ff  a. 

The   constant   of   integration  is   zero,  because  when  -^  z=  a,  we    have 
/  =  0. 


MECHANICS    OF    SOLIDS.  251 

Making  -j-  =  —  a,  we  have 

which  gives  the  time  of  one  entire  oscilhition,  and  from  wliich  we 
conclude  that  the  oscillations  of  the  same  pendulum  will  be  isochro- 
nal, no  matter  what -the  lengths  of  the  arcs  of  vibration,  provided 
they  be  small. 

If  the  number  of  oscillations  performed  iu  a  given  interval,  say 
ten  or  twenty  minutes^  be  counted,__^ the  duration  of  a  single  oscillation 
will  be  found  by  dividing  the  whole  interval  by  this  number. 

Thus,  let  &  denote  the  time  of  observation,  and  -iV  the  number  of 
oscillations,  then  will 


and  if  the  same  pendulum  be  made  to  oscillate  at  some  other  location 
during  the  same  interval  5,  the  force  of  gravity  being  different,  the 
number  N'  of  oscillations  will  be  different ;  but  we  shall  have,  as 
before,  g'  being  the  new  force  of  gravity. 


—  =  *  •  \  / • 

N'  \      e.  q' 


Squaring   and  dividing  the  first  by  the  second,  we  find 

^=^- (374)' 

that  is  to  say,  the  intensities  of  the  force  of  gravity,  at  different 
places,  are  to  each  other  as  the  squares  of  the  number  of  oscilla- 
tions performed  in  the  same  time,  by  the  same  pendulum.  Hence, 
if  the  intensity  of  gravity  at  one  station  be  known,  it  will  be  easy 
to  fmd  it  at  others. 

§  231. — From  Equation    (372),  we  have 

-51  .2»ir2  =  2  J/.y.e(cos-|  —  cosa);     .     .     (375) 


252  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and  making  * 

—^  =   ^1 ;    2  m  r^  z^  I\    e  (cos  n}/  —  cos  a)  =  IT-, 
d  t 

we  have 

/.  F,2  -  2  3I.g.H; (376) 

in  which  H,  denotes  the  vertical  height  passed  over  by  the  centre 
of  gravity,  and  from  which  it  appears  that  the  pendulum  will  come 
to  rest  whenever  -^^  becomes  equal  to  a,  on  either  side  of  the  ver- 
tical  plane  through   the   axis. 

§  232. — If  the  whole  mass  of  the  pendulum  be  conceived  to  be 
concentrated  into  a  single  point,  the  centre  of  gravity  must  go 
there  also,  and  if  this  point  be  connected  with  the  axis  by  a  medium 
without  weight,  we  have  what  is  called  a  simple  pendulum.  Deno- 
ting the  distance  of  the  point  of  concentration  from  the  axis  by  I, 
we  have 

^^  =  0 ;     e  —  I, 

which  reduces  Equation  (374)  to 

t  =  *-\A (377) 


9 


If  the  point  be  so  chosen  that 


n/^-v/^ 


+    C2 


e.g 
or, 

Z.  2      I      p2 

^=^f^;     •        (3T8) 

the  simple  and  compound  pendulum  will  perform  their  oscillations  in 
the  same  time.  The  former  is  then  called  the  equivalent  simple  pen- 
dulum;  and  the  point  of  the  compound  pendulum  into  which  the 
mass  may  be  concentrated  to  satisfy  this  condition  of  equal  duration, 
is  called  the  centre  of  oscillation.  A  line  through  the  centre  of 
oscillation  and  parallel  to  the  axis  of  suspension,  is  called  an  axis  of 
oscillation. 


MECHANICS     OF    SOLIDS.  253 

§233. — The  axes  of  oscillation  and  of  suspension  are  reciprocal. 
Denote  the  length  of  the  equivalent  simple  pendulum  when  the  com- 
pound pendulum  is  inverted  and  suspended  from  its  axis  of  oscillation, 
by  /' ,  and  the  distance  of  this  latter  axis  from  the  centre  of  gravity 
by  e'  then  will 

I'  =  e  -\-  e'     or     e'  =  I  —  e; 

and,  Equation  (378), 

_  k,^  +  e'^    _  k,^  +  {I-  ey 
''   -         e'         ~  l-e 

and  replacing  ?,  by  its  value  in  Equation  (378),  we  find 
sii—  e 

■e 

That  is,  if  the  old  axis  of  oscillation  be  taken  as  a  new  axis  of  sus- 
pension, the  old  axis  of  suspension  becomes  the  new  axis  of  oscilla- 
tion. Tliis  furnishes  an  easy  method  for  finding  the  length  of  an 
equivalent  simple   pendulum. 

Differentiating  Equation  (378),  regarding  I  and  e  as  variable,  we 
have 

8A  _    e^  -  k;^  . 

and  if  ?  be   a   minimum, 

whence, 

e  =  ±k^. 

But  when  ^  is  a  minimum,  then  will  t  be  a  minimum,  Equa- 
tion (377).  That  is  to  say,  the  time  of  oscillation  Avill  be  a 
minimum  when  the  axis  of  suspension  passes  through  the  principal 
centre  of  (/>/ration,  and  the  time  will  be  longer  in  proportion  as  the 
axis  recedes  from    that   centre. 


254 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


^C^ 


Let  A  and  A'  be  two  acute  parallel  prismatic  axes  firmly  con- 
nected with  the  pendulum,  the  acute  edges 
beinc:  turned  towards  each  other.  The 
oscillation  may  be  made  to  take  place 
about  either  axis  by  simply  inverting  the 
pendulum.  Also,  let  Jif  be  a  sliding  mass 
capable  of  being  retained  in  any  position 
by  the  clamp-screw  H.  For  any  assumed 
position  of  M,  let  the  principal  radius  of 
gyration  be  GO;  with  6^  as  a  centre, 
G  C  an  radius,  describe  the  circumference 
CSS'.  From  what  has  been  explained, 
the  time  of  oscillation  about  either  axis 
will   be   shortened    as    it    approaches,   and 

lengthened  as  it  recedes  from  this  circumference,  being  a  minimum, 
or  least  possible,  when  on  it.  By  moving  the  mass  M,  the  centre 
of  gravity,  and  therefore  the  gyratory  circle  of  which  it  is  the 
centre,  may  be  thrown  towards  either  axis.  The  pendulum  bob  being 
made  heavy,  the  centre  of  gravity  may  be  brought  so  near  one  of 
the  axes,  say  A',  as  to  place  the  latter  within  the  gyratory  cir- 
cumference, keeping  the  centre  of  this  circumference  between  the 
axes,  as  indicated  in  the  figure.  In  this  position,  it  is  obvious  that 
any  motion  in*  the  mass  M  would  at  the  same  time  either  shorten 
or  lengthen  the  duration  of  the  oscillation  about  both  axes,  bi^t 
unequally,  in  consequence  of  their  unequal  distances  from  the  gyratory 
circumference. 

The  pendulum  thus  arranged,  is  made  to  vibrate  about  each  axis 
in  succession  during  equal  intervals,  say  an  hour  or  a  day,  and  the 
number  of  oscillations  carefully  noted;  if  these  numbers  be  the 
same,  the  distance  between  the  axes  is  the  length  /,  of  the  equiva- 
lent simple  pendulum  ;  if  not,  then  the  weight  M  must  be  moved 
towards  that  axis  whose  number  is  the  least,  and  the  trial  repeated 
till  the  numbers  are  made  equal.  The  distance  between  the  axes 
may  be   measured  by  a  scale  of  equal  parts. 

§  234. — From  this  value  of  /,  we  may  easily  find  that  of  the  simple 
second's   pendulum ;    that  is  to   say,  the  simple  pendulum  which  will 


MECHANICS    OF     SOLIDS.  255 

perform  its  vibration  in  one  second.  Let  iV,  be  the  number  of 
vibrations  performed  in  one  hour  by  the  compound  pendulum  whose 
equivalent  simple  pendulum  is  /;  the  number  performed  in  the 
same  time  by  the  second's  pendulum,  whose  length  we  will  denote 
by  l\  is  of  course  3G00,  being  the  number  of  seconds  in  1  hour, 
and  hence, 


N                    \ 

^   9 

=  r  =  *  X 
3600^                       \ 

r- 

'    9 

and  because  the  force  of  gravity  at  the  same  station  is  constant, 
we  find,  after  squaring  and  dividing  the  second  equation  by  the  first, 

V  =  _llll_ (379) 

Such  is,  in  outline,  the  beautiful  process  by  which  Kater  determined 
the  length  of  the  simple  second's  pendulum  at  the  Tower  of  London 
to   be    39,13908  inches,  or  3,20159  feet. 

As  the  force  of  gravity  at  the  same  place  is  not  supposed  to 
change  its  intensity,  this  length  of  the  simple  second's  pendulum 
must  remain  forever  invariable ;  and,  on  this  account,  the  English 
have  adopted  it  as  the  basis  of  their  system  of  weights  and  measures. 
For  this  purpose,  it  was  simply  necessary  to  say  that  the  3,2^150"' 
part  of  the  simple  second's  pendulum  at  the  Toiver  of  London  shall 
be  one  English  foot,  and  all  linear  dimensions  at  once  result  from 
the  relation  they  bear  to  the  foot;  that  the  gallon  shall  contain 
_2Ayh  of  a  cui^ic  f^^ot^  and  all  measures  of  volume  are  fixed  by  the 
relations  which  other  volumes  bear  to  the  gallon;  and  finally,  that 
a  cubic  foot  of  distilled  water  at  the  temperature  of  sixty  degrees 
Tahr.  shall  weigh  one  thousand  ounces,  and  all  weights  are  fixed  by 
the   relation   they   bear   to    the  ounce. 

§235.— It  is  now  easy  to  find  the  apparent  force  of  gravity  at 
London  ;  that  is  to  say,  the  force  of  gravity  as  afiected  by  the  cen- 
trifuo-al  force  and  the  obhitencss  of  the  earth.     The  time  of  oscillation 


256  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

being   one  second,  and   the   length  of  the   simple   pendulum   3,26159 
feet.  Equation  (377)  gives 


/3,26159 

1    =  'K\/   ■ ; 

whence, 

y  =  7r2  (3,26159)  =  (3,1416)2  ^  (3,26159)  =  32,1908  feet. 

From  Equation  (377),  Ave   also  find,  by  making    t  one   second. 


and  assuming 


we  have 


I  ^=  X  -T  y  cos  2  4', 


A  =  a;  +  ycos24. (380)  . 

Now  starting  with  the  value  for  g  at  London,  and  causing  the 
'  same  pendulum  to  vibrate  at  places  whose  latitudes  are  known,  we 
obtain,  from  the  relation  given  in  Equation  (374)',  the  corresponding 
values  of  g,  or  the  force  of  gravity  at  these  places ;  and  these 
values  and  the  corresponding  latitudes  being  substituted  successively 
in  Equation  (380),  give  a  series  of  Equations  involvmg  but  two  un- 
known quantities,  which  may  easily  be  found  by  the  method  of 
least   squares. 

In  this   way  it   has   been  ascertained  that 

•K-^.x  =  32,1803     and     ^^y  =  -  0,0821  ; 
V  hence,  generally, 

g  =  32,1803  -  0,0821  cos  2  4. ;      ....     (381) 

and   substituting   this   value    in    Equation    (377),  and    making   ^  =  1, 

we  find 

/ 
^  m  3,26058  —  0,008318  cos  2  4     ....     (382) 

Such   is   the   length  of   the   simple   second's    pendulum   at   any   place 
of  which   the   latitude  is  4. 


MECHAXICS     OF    SOLIDS, 


257 


If  we    make   4.  =1  40°  42'  40",  the    latitude   of  the    City  Hall  of 
New  York,  we   shall  find 


I  =  3,25938 


39,11256. 


§230. — The  principles  which  have  just  been  explained,  enable  us 
to  find  the  moment  of  ipfcrtia  of  any  body  turning  about  a  fixed 
axis,  with  great  accuracy,  no  matter  what  its  figure,  density,  or  the 
distribution  of  its  matter.  If  the  axis  do  not  pass  through  its  centre 
of  gravity,  the  body  will,  when  deflected  from  its  position  of  equi- 
librium, oscillate,  and  become,  in  fact,  a  compound  pendulum  ;  and 
denoting  the  length  of  its  equivalent  simple  pendulum  by  I,  we  have, 
after   multiplying  Equation  (378)  by  M, 


M.l.e  =  M  [k,^  +  e2)  _  2  m  ?-2 


•(383) 


9 


W 

—  •  I  ,e 
9 


(384) 


in  which    W  denotes    the   weight  of  the   body. 

Knowing  the  latitude  of  the  place,  the  length  V  of  the  simple 
second's  pendulum  is  known  from  Equation  (382)  ;  and  counting  the 
number  N  of  oscillations  performed  by  the  body  in  one  hour. 
Equation  (379)    gives 


I  ^ 


V  •  (3600)2 


To  find  the  value  of  f,  which  is 
the  distance  of  the  centre  of  gravity 
from  the  axis,  attach  a  spring  or 
other  balance  to  any  point  of  the 
body,  say  its  lower  end,  and  bring 
the  centre  of  gravity  to  a  horizontal 
plane  through  the  axis,  which  posi- 
tion will  be  indicated  by  the  max- 
imum reading  of  the  balance.  De- 
noting by  a,  the  distance  from  the  axis   C  to  the  point  of  support  R. 

17 


258  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

and  by  6,  the  maximum  indication  of  the  balance,  we  have,  from 
the  principle  of  moments, 

ba  -  We. 

The  distance  a,  may  be  measured  by  a  scale  of  equal  parts.  Sub- 
stituting the  values  of  TF,  e  and  I  in  the  expression  for  the  moment 
of  inertia.  Equation  (38-4),  we   get 

"-.^l^^^I. ,3S5) 

If  the  axis  pass  through  the  centre  of  gravity,  as,  for  example, 
in  the  fiy-ivheel,  it  will  not  oscillate ;  in  which  case,  take  Equation 
(383),«  from  which  we  have 

i¥/t,2  =  M.l.e  -  J/e2. 

Mount  the  body  upon  a  parallel  axis  A,  not  passing  through  the  cen- 
tre of  gravity,  and  cause  it  to  vibrate 
for  an  hour  as  before;  from  the  num- 
ber of  these  vibrations  and  the  length 
of  the  simple  second's  pendulum,  the 
value  of  /  may  found ;  M  is  known, 
being  the  weight  W  divided  by  g  ;  and 
e  may  be  found  by  direct  measure- 
ment, or  by  the  aid  of  the  spring 
balance,  as  already  indicated;  whence  Jr,  becomes  known. 


MOTION     OF   A   BODY   ABOUT   AN     AXIS    UNDER    THE   ACTION    OF    niPUL- 

SIVE    rOKCES. 

g  237. — If  the  forces  be  impulsive,  we  may,  §  184,  replace  in 
Equation  (366)  the  second  differential  co-efficients  of  x,  y,  2,  by  the 
first  differential  co-efficients  of  the  same  variables,  which  will  reduce 
it  to 


2  P  (  s  cos  a.  —  X  cos  j)  —2m 


zdx  —  xdz 


0 


MECHANICS     OF    SOLIDS. 


259 


and    replacing    dx,  dy,  dz,    by  their  values   in    Equations    (368),  we 
find 


d-\>        H  P(z  cos  a  —  X  cos  y) 
IT  ~  2mr2 


(386) 


That  is,  the  angular  velocity  of  a  body  retained  by  a  fixed  axis,  and 
subjected  to  the  sinmltaneous  action  of  impulsive  forces,  is  equal  to  the 
sum  of  the  moments  of  the  impressed  forces  divided  by  the  moment  of 
inertia  luith  reference  to  this  axis. 


BALISTIO    PENDULUM. 

§  238. — In  artillery,  the  initial  velocity  of  projectiles  is  ascertained 
by  means  of  the  balistic  pendulum, 
which  consists  of  a  mass  of  matter 
suspended  from  a  horizontal  axis 
in  the  shape  of  a  knife-edge,  after 
the  manner  of  the  compound  pen- 
dulum. The  bob  is  either  made 
of  some  unelastic  substance,  as 
wood,  or  of  metal  provided  with 
a  large  cavity  filled  with  some 
soft  matter,  as  dirt,  which  re- 
ceives the  projectile  and  retains 
the  shape  impressed  upon  it  by  the 
blow 

Denote  by  V  and  m,  the  initial  velocity  and  mass  of  the  ball ; 
V,  the  angular  velocity  of  the  balistic  pendulum  the  instant  after 
the  blow,  /  and  31  its  moment  of  inertia  and  mass.  Also  let  r 
represent  the  distance  of  the  centre  of  oscillation  of  the  pendulum 
from  the  axis  A.  That  no  motion  may  be  lost  by  the  resistance 
of  the  axis  arising  from  a  shock,  the  ball  must  be  received  in  the 
direction  of  a  line  passing  through  this  centre  and  perpendicular  to 
the  line  A  0.      This  condition   being    satisfied,  we  have 


S  F  {z  cos  a  —  X  cos  y)  =  r 

2  7n  ?'2  =  ??i  r^  -f  /; 


V; 


260         ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and  Equation   (386)  becomes 

rm  V 


from   which   we   find 

^^(mr^+/)F,. ^3g^.^ 

■Ill  r 

the  velocity  F,  becomes  known,  therefore,  when  Fj  is  known,  since 
all  the  other  quantities  may  be  easily  found  by  the  methods  already 
explained.  To  find  F,  denote  by  //,  the  greatest  height  to  which 
the  centre  of  gravity  of  the  pendulum  is  elevated  by  virtue  of 
"this  angular  velocity;  then,  since  the  moment  of  inertia  of  the  ball 
is  mr^  §  181,  we  have,  from  the  principle  of  the  living  force,  Equa- 
tion (376), 

(/  +  m  r2)  Fi2  =  2  (  i/  +  m)  g  H-, 

whence, 

(/  +  mr^)F;-^,^_ 
{M+m)g 

Denoting  by  T  the  time  of  a  single  oscillation  of  the  pendulum 
after  it  receives  the  ball,  we  have,  by  multiplying  both  terms  of 
the  fraction  under  the  radical  sign  in  Equation  (374)  by  M  +  m, 
and  reducing  by  the  relation,  (J/  +  m)  {k^^  +  e^)  =  {M  +  7n)P, 
Equation  (244^, 


V  (M  4 


T  ' 


{M+m)B.g 

D  beina:  the  distance  from  the  axis  to  the  centre  of  gravity ;   whencg, 

/+  mr2     _  DT'^^ 
{M  +  7n)  g    ~      TT-    ' 

and  this  value,  substituted  in  the   equation  of  the  living  force,  gives 
Jj  T" 

whence, 

*^'  -  T    V  D   ' 


MECHANICS    OF    SOLIDS, 


261 


also, 


2        (i¥+  m)g.D.T^ 


and  because,  Equation  (377), 


we  find 


T-'ff 


Substituting   these   values  of    F,,  I  +  mr"^  and  r  in  Equation  (387), 
we  find 

I    ^  m 

or,   replacing   the   masses    by   their   values   in   terms  of   the   weights 
and  force   of  gravity, 

W  -{-10 


in  which  W  and  %u  denote  the  weights  of  the  pendulum  and  ball 
respectively. 

Observe  that  H,  is  the  height  to  which  the  centre  of  gravity 
rises  in  describing  the  arc  of  a  circle  of 
which  D  is  the  radius.  Let  G  G'  K  be 
half  of  the  circumference  of  which  this  arc 
is  a  part,  G  and  G'  the  initial  and  termi- 
nal positions  of  the  centre  of  gravity  du- 
ring the  ascent ;  draw  G'  R  perpendicular 
to  K  G.  Then,  because  A  G  =  D,  and 
G  H  =:  II,  we  have,  from  the  property 
of  the  circle, 

EG'  =  ^II{2D  -  H)  ; 

and  if  the  pendulum  be  made  large,  so  that  the  arc  G  G'  shall  be 
very  small,  which  is  usually  the  case,  //  may  be  neglected  in  com- 
parison   with  2  Z>,  and   therefore 

BG'  =  ^2H.D ; 


262  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

^2HD  is  half  the  chord  of  the  arc  described  by  the  centre  of 
gravity  in  one  entire  oscillation.  Denoting  this  chord  by  C,  and 
substituting   above,  we  have 


T 

From  this  equation,  we  may  find  the  initial  velocity  V ;  and 
for  this  purpose,  it  will  only  be  necessary  to  have  the  duration 
of  a  single  oscillation,  and  the  amplitude  of  the  arc  described  by 
the  centre  of  gravity  of  the  pendulum.  The  process,  for  finding 
the  time  has  been  explained.  To  find  the  arc,  it  will  be  suffi- 
cient to  attach  to  the-  lower  extremity  of  the  pendulum  a  pointer, 
and  to  fix  on  a  permanent  stand  below,  a  circular  graduated  groove, 
whose  centre  of  curvature  is  at  A  ;  the  groove  being  filled  with 
some  soft  substance,  as  tallow,  the  pointer  will  mark  on  it  the 
extent  of  the  oscillation.  Knowing  thus  the  arc,  denoted  by  ^,  and 
the  value  of  I>,   found   as   already  described,  §236,  we   have 


whence, 
and  finally, 


(7=  2Z>.sin|-^; 


p.^^.2>.i:±Jfsini^. (388) 


PART    II. 


MECHAIICS    OF    FLUIDS 


INTRODUCTORY     REMARKS. 


^2-VH-   -  ^<^2>^=^V  ^      -  ^Z>  ^-^t^ 


^^-  ^  ~^<^'^t^^2)('  -Jf-^t^j  ^^7^..^B^^^^//-      upon    the 

he    attrac- 
eld  firmly 

J'-—  j.^d-T)    yVf H^        r,-    ,    ^  irence  be- 

;ofter,  and 
its  tigure  yields  more  readily  to  external  pressure.  >v  hen  these 
forces  are  equal,  the  particles  will  yield  to  the  slightest  force,  the 
body  will,  under  the  action  of  its  own  weight,  and  the  resistance 
of  the  sides  of  a  vessel  into  Svhich  it  is  placed,  readily  take  the 
figure  of  the  latter,  and  is  liquid.  Finally,  when  the  repulsive  ex- 
ceed the  attractive  forces,  the  elements  of  the  body  tend  to  separate 
from  each  other,  and  require  either  the  application  of  some  extra- 
neous force  or  to  be  confined  in  a  closed  vessel  to  keep  them 
together ;  the  body  is  then  a  gas.  fn  the  vast  range  of  relation 
among  the  molecular  forces,  from  that  which  distinguishes  a  solid  to 
that  which  determines  a  gas  or  vapor,  bodies  are  found  in  all  possible 
conditions — solids  run  imperceptibly  into  liquids,  and  liquids  into 
gases.  Hence  all  classification  of  bodies  founded  on  their  physical 
properties   alone,  must,  of  necessity,  be  arbitrary. 

1 240. — Any    body    whose    elementary    particles    admit    of   motion 


262  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

■J'2IID  is  half  the  chord  of  the  arc  described  by  the  centre  of 
gravity  in  one  entire  oscillation.  Denoting  this  chord  by  C,  and 
substituting   above,  we  have 

*  W  4-  w 


T 

From  this  equation,  we  may  find  the  initial  velocity  V ;  and 
for  this  purpose,  it  will  only  be  necessary  to  have  the  duration 
of  a  single  oscillation,  and  the  amplitude  of  the  arc  described  by 
the  centre  of  gravity  of  the  pendulum.  The  process  for  finding 
the  time  has  been  explained.  To  find  the  arc,  it  will  be  suffi- 
cient to  attach  to  the-  lower  extremity  of  the  pendulum  a  pointer, 
and  to  fix  on  a  permanent  stand  below,  a  circular  graduated  groove, 
whose  centre  of  curvature  is  at  A  ;  the  groove  being  filled  with 
some  soft  substance,  as  tallow,  the  pointer  will  mark  on  it  the 
extent  ''        '       ■^^""   *^'^   ^^^    denoted  by  ^,  and 

the  va 


whenc( 


and   finally, 


F  =  ^.i).^^^sinia. (388) 


PART    II. 


MECHANICS    or    FLUIDS 


INTRODUCTORY     REMARKS. 

§239. — The  physical  condition  of  every  body  depends  upon  the 
relation  subsisting  among  its  molecular  forces.  When  the  attrac- 
tions prevail  greatly  over  the  repulsions,  the  particles  are  held  firmly 
together,  and  the  body  is  solid.  In  proportion  as  the  difference  be- 
tween these  two  sets  of  forces  becomes  less,  the  body  is  softer,  and 
its  figure  yields  more  readily  to  external  pressure.  When  these 
forces  are  equal,  the  particles  will  yield  to  the  slightest  force,  the 
body  will,  under  the  action  of  its  own  weight,  and  the  resistance 
of  the  sides  of  a  vessel  into  Svhich  it  is  placed,  readily  take  the 
figure  of  the  latter,  and  is  liquid.  Finally,  when  the  repulsive  ex- 
ceed the  attractive  forces,  the  elements  of  the  body  tend  to  separate 
from  each  other,  and  require  either  the  application  of  some  extra- 
neous force  or  to  be  confined  in  a  closed  vessel  to  keep  them 
together ;  the  body  is  then  a  gas.  In  the  vast  range  of  relation 
among  the  molecular  forces,  from  that  which  distinguishes  a  solid  to 
that  which  determines  a  gas  or  vapor,  bodies  are  found  in  all  possible 
conditions — solids  run  imperceptibly  into  liquids,  and  liquids  into 
gases.  Hence  all  classification  of  bodies  founded  on  their  physical 
properties   alone,  must,  of  necessity,  be  arbitrary. 

g240. — Any    body    whose    elementary    particles    admit    of   motion 


264  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

among  each  other,  is  called  a  Jluid — such  as  water,  wine,  mercury, 
the  air,  and,  in  general,  liquids  and  gases  ;  all  of  which  are  distin- 
guished from  solids  by  the  great  mobility  of  their  particles  among 
themselves.  This  distinguishing  property  exists  in  different  degrees 
in  different  liquids — it  is  greatest  in  the  ethers  and  alcohol ;  it  is 
less  in  water  and  wine ;  it  is  still  less  in  the  oils,  the  sirups, 
greases,  and  melted  metals,  that  flow  with  difficulty,  and  rope  when 
poured  into  the  air.  Such  fluids  are  said  to  be  viscous,  or  to  possess 
viscosity.  Finally,  a  body  may  approach  so  closely  both  a  solid  and 
liquid,  as  to  make  it  difficult  to  assign  it  a  place  among  either 
class,  as  paste,  fidty,  and   the   like. 

8241. — Fluids  are  divided  in  mechanics  into  two  classes,  viz.: 
compressible  and  incompressible.  The  term  incompressible  cannot,  in 
strictness  of  propriety,  be  applied  to  any  body  in  nature,  all  being 
more  or  less  compressible;  but  the  enormous  power  required  to 
change,  in  any  sensible  degree,  the  volumes  of  liquids,  seems  to 
justify  the  term,  when  applied  to  them  in  a  restricted  sense.  The 
ffases  are  highly  compressible.  All  liquids  will,  therefore,  be  regarded 
as   incompressible ;    the  yuses  as    compressible. 

§242. — The  most  important  and  remarkable  of  the  gaseous  bodies 
is  the  atmosphere.  It  envelops  the  entire  earth,  reaches  far  beyond 
the  tops  of  our  highest  mountains,  and  pervades  every  depth  from 
which  it  is  not  excluded  by  the  presence  of  solids  or  liquids.  It 
is  even  found  in  the  pores  of  these  latter  bodies.  It  plays  a  most 
important  part  in  all  natural  phenomena,  and  is  ever  at  work  to 
influence  the  motions  within  it.  It  is  essentially  composed  of  oxrjgen 
and  nitrogen,  in  a  state  of  mechanical  mixture.  The  former  is  a 
supporter  of  combustion,  and,  with  the  various  forms  of  carbon,  is 
one  of  the  principal  agents  employed  in  the  development  of  mechan- 
ical power. 

The  existence  of  gases  is  proved  by  a  multitude  of  facts.  Con- 
tained in  an  inflexible  and  impermeable  envelope,  they  resist  pressure 
like  solid  bodies.  Gas,  in  an  inverted  glass  vessel  plunged  into 
water,  will  not  yield  its  place  to  the  liquid,  unless  some  avenue  of 
escape   be   provided  for   it.     Tornadoes  which  uproot   trees,  overturn 


MECIIAXICS     OF     FLUIDS.  265 

houses,  and  devastate  entire  districts,  are  but  air  in  motion.  Air 
opposes,  by  its  inertia,  the  motion  of  other  bodies  through  it,  and 
this  opposition  is  called  its  resistance.  Finally,  we  know  that  wind 
is  employed  as  a  motor  to  turn  mills  and  to  give  motion  to  ships 
of  the  largest   kind. 

§  243. — In  the  discussions  which  are  to  follow,  fluids  will  be  con- 
sidered as  without  viscosity ;  that  is  to  say,  the  particles  will  be 
supposed  to  have  the  utmost  freedom  of  motion  among  each  other. 
Such  fluids  are  said  to  be  perfect.  The  results  deduced  upon  the 
hypothesis  of  perfect  fluidity  will,  of  course,  require  modification 
when  applied  to  fluids  possessing  sensible  viscosity.  The  nature  and 
extent  of  these  modifications   can  be    known   only  from  experiments. 


MAEIOTTE  S   LAW. 

§244. — Gases  readily  contract  into  smaller  volumes  when  pressed 
externally  ;  they  as  readily  expand  and  regain  their  former  dimen- 
sions when  the  pressure  is  removed.  They  are  therefore  both  com- 
pressible and  elastic. 

It  is  found  by  experiment,  that  the  change  in  volume  is,  for  a 
constant  temperature,  always  directly  proportional  to  the  change  of 
pressure.  The  density  of  the  same  body  is  inversely  proportional  to 
the  volume  it  occupies.  If,  therefore,  P  denote  the  pressure  upon 
a  unit  of  surface  which  will  produce,  at  a  given  temperature,  say 
32°  Fahr.,  a  density  equal  to  unity,  and  D  any  other,  density,  and 
p  the  pressure  upon  a  unit  of  surface  which  will,  at  the  same  tem- 
perature of  the  gas,  produce  this  density,  then,  according  to  the  ex- 
periments above   referred   to,  will 

p  =  P.D (389) 

This  law  was  investigated  by  Boyle  and  ]\rariotte,  and  is  known 
as  Mariotte's  Law.  By  experiments  made  at  Paris,  it  was  found- that 
this  law  obtains,  when  air,  in  its  ordinary  condition,  is  condensed  27 
and.  rarefied   112   times. 


26G  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


LAW   OF  THE    PKESSHRE,   DENSITY,    AND    TEMPEKATUKE. 

8  245. — It  is  a  universal  law  of  nature  that  heat  expands  all 
bodies,  and   is    ever   active   in  producing    changes  of  density. 

It  has  been  ascertained,  experimentally,  that  air,  subjected  to  any 
constant  pressure,  will  alter  its  volume  by  0,00208*''  part  of  that 
which  it  has  at  32°  Fahr.,  for  each  degree  of  the  same  scale  above 
and  below  this  temperature ;  so  that  if  Fj  be  the  volume  of  the 
air  at  32°,  and   V  its  volume  at   any  other  temperature  t,  then  will 

V  =  V,[\  +  {t  -  32°)  0,00208]     ....     (390) 

If  D^  be  the  density  at  32°,  under  a  pressure  p,  and  D  that  at  the 
temperature  t,  under  the  same  pressure,  then,  because  the  densities 
are   inversely  as   the  volumes,  will 

Fi  :  F,  [1  +  (^  -  32°)  0,00208]  :  :  D  :  I>r, 

whence, 

J)  =  ^ (391) 

1  +  (<  -  32°) .  0,00208  ^       ^ 

If  2^/  denote  the  pressure  necessary  to  restore  this  air  to  the  density 
D^,  we  shall   have,  from  Mariotte's  law, 

:  D,  ::  p:  p,  \ 


1  +  (/  -  32°)  0,00208 


whence 


p^  =p\\  +  {t  -  32°)  0,00208].     .     .     .     •    (392) 


Let  the  pressure  p,  be  produced  by  the  weight  of  a  column  of 
mercury,  having  a  base  unity,  and  an  altitude  A^^,  taken  at  a  given 
latitude,  say  that  of  45°,  and  denote  the  density  of  the  mercury  at 
32°  Fahr.,  by  D^\    its   weight   will  be 

p  =  DJi,,g'', 

in   which  g'   denotes   the   force   of  gravity    at   the  latitude   of  45°. 
Substituting  this   for  p^  in  Equation  (389),  we  have 

BJi.g'^PD- 


MECHANICS    OF    FLUIDS.  267 

whence, 

D       ' 

and   substituting  the  value  of  D,  given  in  Equation  (391),  this  becomea 

P  =  -^' —  [1  +  (^  -  32°)  0,00208].     .     .     .  (393) 
From  Equation    (389),  Ave  have 

and  substituting   the   value   for   P   above,  we   get 


D  = 


JJ,„  A,,  y'  [1  -f-  {(  -  32°)  0,0020SJ 

Denote  by  h,  the  height  of  the  column  of  mercury  at  t°  necessary 
to  produce  upon  a  unit  of  surface  the  pressure  ]),  then  D',„  deno- 
ting  the   corresponding  density   of  the    mercury,  will 

which  substituted  for  2^  above,  gives,  after  striking  out  the  common 
factors, 

B^ ^ 

~  h,,  [!  +  (<-  32°)  0,00208]  "  I)„  ' 

From  the  experiments  of  Petit  and  Dulong,  it  is  found  that  mer- 
cury expands  g^g-Q  part  of  its  volume  for  each  degree  of  Fahren- 
heit's scale  by  which  its  temperature  is  increased,  and  that  it  con- 
tracts according  to  the  same  law  as  its  temperature  is  diminished. 
If,  therefore,  T  denote  the  standard  temperature,  and  T'  the  temper- 
ature of  observation ;  h^^  the  altitude  which  the  barometer  would 
have  indicated  at  the  standard  temperature,  and  h  the  observed  alti- 
tude, then    will, 

''"  =  ^'  [l  +  ^T)^]  =  ni  +  (r  -  r) .  0,0001001].  (394) 

But  because  the  mass  of  mercury  to  exert  the  same  pressure 
must   be   the  same,  we   have 

D',„ .  h  =  D„,  h^„ 


268  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


^'.  J>u 


Dr.  h 

"which   substituted    above,  gives 


=  1  +  (^-  T').  0,0001001; 


D  ^  D 


h    1  +  (y-  T) .  0,0001001 
'    h^/   1  +  (i  —  32°yTo700208 


(395) 


In  which,  if  A^,  =  30'",  and  T  —  32°,  then  will  D^  become  the  tabu- 
lar  density.     Table  (000). 


EQUAL    TKAN8MISSI0N    OF    PKESSUEE. 

§246. — Let  E H  L,  represent  a  closed  vessel  of  any  shape,  with 
which  two  piston  tubes  A  B'  and 
D  C  communicate,  each  tube  be- 
ing provided  with  a  piston  that 
fits  it  accurately  and  which  may 
move  within  it  with  the  utmost 
freedom.  The  vessel  'being  filled 
Avith  any  fluid,  let  forces  P  and 
P',  be  apjjlied,  the  former  per-  . 
pendicularly  to  the  piston  A  B, 
and  the  latter  in  like  direction 
to  the  piston  CD,  and  suppose 
these    forces    in    equilibrio,    which 

they  may  be,  since  the  fluid  cannot  escape.  Now  let  the  piston 
A  B  be  moved  to  the  position  A'  B' ;  the  piston  CD  will  take 
some  new"  position,  as  C  D' .  And  denoting  by  s  and  s',  the  dis- 
tances A  A'  and  C  C",  respectively,  we  have,  from  the  principle  of 
virtual  velocities, 

Ps  =  P'  s'. 

Denote  the  area  of  the  piston  A  B  by  a,  and  that '  of  the  piston 
C  D  hy  a',  then  will  the  volume  of  the  fluid  Avhich  was  thrust  from 
the   tube  A  B\  be  measured  by  « ,  s,  and  that  which  entered  the  tube 


MECHANICS     OF    FLUIDS.  269 

D  C",  will  be  measured  by  a'  s'.  But  the  pressure  upon  the  pistons 
and  the  temperature  remaining  the  same,  the  entire  volume  of  the 
fluid   in   the   vessel   and   tubes   will   be   unchanged.      Hence, 

a  s  =  a'  s' ;  ^ 

dividing  the  equation  above   by  this  one,  we   have 


P   _  P^ 

a  a' 


(39G) 


That  is  to  say,  two  forces  applied  to  pistons  which  communicate  freely 
with  each  other  through  the  intervention  of  some  confined  fluid,  will 
he  in  equilibrio  when  their  intensities  are  directly  proportional  to  the 
areas    of  the  2}ist07is  upon  tvhich  they  act. 

This  result  is  wholly  independent  of  the  relative  dimensions  and 
positions  of  the  pistons ;  and  hence  we  conclude  that  any  pressure 
communicated  to  one  or  more  elements  of  a  fluid  mass  in  equilibrio,  is 
equally  transmitted  throughout  the  whole  fluid  in  every  direction.  This 
law  which  is  fully  confirmed  by  experiment,  is  known  as  the  prin- 
ciple of  equal  transmission  of  pressure. 

§247. — Let  a  become  the  superficial  unit,  say  a  square  inch  or 
square  foot,  then  will  P  be  the  pressui-e  applied  to  a  unit  of  sur- 
face, and.  Equation  (396), 

P'  =  Pa'. (397) 

That  is,  the  pressure  transmitted  to  any  portion  of  the  surface  of 
the  containing  vessel,  will  be  equal  to  that  applied  to  the  unit  of 
surface  multiplied  by  the  area  of  the  surface  to  which  the  transmis- 
sion  is  made. 

§  248 — Since  the  elements  of  the  fluid  arc  supposed  in  equilibrio, 
the  pressure  transmitted  to  the  surface  through  the  elements  in  con- 
tact with  it,  must,  §217  and  Equations  (332),  be  normal  to  the  sur- 
face. That  is,  the  2^>'essure  of  a  fluid  against  any  surface,  acts  always 
in    the   direction    of  the    normal. 


270  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


MOTION   OF   THE   FLUID   PARTICLES. 

§249. — The  particles  of  a  fluid  having  the  utmost  freedom  of 
motion  among  one  another,  all  the  forces  applied  at  each  particle 
must  be  in  equilibrio.  Regarding  the  general  Equation  (40)  as  ap- 
plicable to  a  single  particle,  whose  co-ordinates  are  x,  y,  2,  we  shall 
have 

X  -  x^,     y  =  Vn     ^  =  2y  J 

and  supposing  the   particle   to   have   simply  a   motion  of  translation, 
we  also  have 

5(p  =0;     54.  =  0;     Svi  =  0; 

and   that   equation   becomes 

(2  P  cos  a  —  m  •  -j-y)  ^  X 
+    (nFcoslS  -ni'^^  Sy    [   =  0  ; 


whence,  upon   the   principle  of  indeterminate   co-efficients, 


2  P  cos  a  —  m  •  -j-^  =  0  ; 

d^  y 
IP  cos  13  —  m  '  -~  =  0  ; 

d'^  z 
2  P  cos  7  -  »i  •  -j^  =  0. 


(398) 


Now  the  terms  2  P  cos  a,  2  P  cos  /3  and  2  P  cos  7,  are  each  composed 
of  two  distinct  parts,  viz. :  1st.,  the  component  of  the  resultant  of 
the  forces  applied  directly  to  the  particle;  and  2d.,  the  component 
of  the  pressure  transmitted  to  it  from  a  distance,  arising  from  the 
forces  impressed   upon    other   particles. 

Denote  by  X,   T  and  Z,  the  accelerations,  in  the  directions  of  the 
axes   X,  y  z,   respectively,    due  to   the  forces   applied  directly  to   the 


MECHANICS     OF     FLUIDS. 


271 


particle ;  then  m,  being  the  mass   of  the  particle,  the  components  of 
the  forces  directly  impressed  will  be 

mX ;     m  Y;     mZ. 

The  pressure  transmitted  will  depend  upon  the  particle's  place, 
and  will  be  a  function  of  its  co-ordinates  of  position.  Denote  by  ^, 
the  pressure  upon  a  unit  of  surface,  on  the  supposition  that  every 
point  of  the  unit  sustains  a  pressure  equal  to  that  communicated  to 
the    particle  from  a   distance;    then  will 

J)  =  F  (x,  y,  z). 

Conceive  each  particle  of  the  fluid  to  consist  of  a  small  rectan- 
gular parallelopipedon  whose 
faces  arc  parallel  to  the  co- 
ordinate planes,  and  whose  con- 
tiguous edges  at  the  time  t^ 
are  dx^  dy  and  dz;  and  let 
.r,  y,  z,  be  the  co-ordinates  of 
the  molecule  in  the  solid  an- 
gles nearest  the  origin  of  co- 
ordinates. Then  would  the 
difference   of   pressure    on    the  /y. 

opposite  faces,  which  are  paral- 
lel  to  the   plane   sy,  were   these  faces   equal   to   unity,  be 

dp 


z 

f 

g 

v 

p?" 

J-' 

/ 
/ 

F{x^  dx,  y,z,)  -  F  (.r,  y,  z,)  =—  •  dx', 

and   upon  the   actual   faces   whose   dimensions    arc   each   dz.dy,  this 
difference   becomes,  Equation  (397), 

dp 


d  X 


d X'd y  '  d z. 


In  like  manner  will  the  difference  of  the  pressures  transmitted 
to  the  opposite  taces  parallel  to  the  planes  zx  and  .ry,  be,  respec- 
tively, 


dp 
dy 


d  y  •  d  z  -  d  X, 


dp 
and     —7^  -dz-  d  X  'dy. 
dz  -^ 


272 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


These  pressures  being  normal  to  the  surfaces  to  which  they  are 
respectively  applied,  they  will  act,  the  first  in  the  direction  of  x, 
the  second  in  the  direction  of  y,  and  the  third  in  the  direction 
of  z.  And  as  these  differences  alone  determine  that  portion  of  the 
motion    due  to    the   transmitt<;d    pressures,  we    have 

2  P  cos  a  =z  mX ; —  •  dx  .dy  .dz\ 

dx 


2  P  cos  /3 


m  Y ; —  '  dy .  dx .  dz  : 

dy        ^ 

_                        „          dj) 
2  P cos  y  =^  m  Z —  •  dz  .  d x  .  dy. 


Denote  by  D  the  density  of  the  mass  ???,  then  will,  Equation  (1)', 
m  ^=:^  D  .  d  x  .  d  y  .  d  z, 
and   by  substitution,  Equations  (398)  become 


1 

dp 
dx 

d^x 
-^         d  {^  ' 

1 
1) 

dp 
dy 

df  ' 

1 

dp 

dz 

-  ^  ~  dW  ' 

(399) 


Denote  by  m,  v  and  w^  the  velocities  of  the  molecule  whose  co- 
ordinates are  xyz,  parallel  to  the  axes  a",  y,  z,  respectively,  at  the 
time  t.  Each  of  these  will  be  a  function  of  the  time  and  the  co- 
ordinates of  the  molecule's  place;  and,  reciprocally,  each  co-ordinate 
will  be  a  function  of  t,  m,  v  and  w  ;  whence.  Equations  (12)  and  (13), 

d"^  X        du        /d  u\     dt       du    dx       du     dy       du     dz 
'df  ~'dt~   \dl/   "eft       d^'dl        cfy  '  I't       'dz  '  dTt  ' 

,     .        dx     dy     dz      .        ,    .         ^  .     , 

and  replacing  -p'    7"'   -^'     hy  their  values  u,  v,  to,  respectively,  we 

have 


d^x 


=  (—) 
\dtJ 


du  du 

-i-  ^  •  «  +  -— 
ax  d  y 


du 

V  +  -- —  •  w\ 
dz 


d  V  d  V 

+    -7—  •  V   +    -r--W, 

dy  dz 


MECHANICS     OF     FLUIDS, 
in   the   same  way, 

d  t-  \  dt  /  dx 

d'^z         /dio\         dw  ^     dw  _     d  to 

=    ( )   H ; —  •  u  -i ; —  •  V  + 

dt"         \  dt  ■^         dx  dy 

which,  substituted  in   Equations  (399),  give 

d  u  d  u  d  u 


273 


dz 


D 


du 

U ; •  V 

:x  dy 


i_   £P. 
l)"d^ 

J_    dj) 
B 


d  V 
dy 

dw 


dz 
dv 


D"dx  ~             \dt/  d 

/dv\  dv 

Y  —  { I -, —  u  — 

\dt/  dx 

dp         „        /dw\  dto               dw               dio 

•  —r-  =  Z  —  \  —f—  } J—  •  ■^ J '" T—  •  ^• 

d  z                   \  d  t  y'  d  x                dy                dz 


V  —  —^ —  -w 
dz 

div 


(400) 


Here  are  three  equations  involving  five  unknown  quantities,  viz,  : 
w,  V,  w,  p  and  D,  which   are   to  be  found  in   terms  of  .r,  y,  z  and  t. 

Two  other  equations  may  be  found  from  these  considerations,  viz  : 
the  velocity  in  the  direction  of  x,  of  the  molecule  whose  co-ordinates 
are  x  y  z,  is  w ;  the  velocity  of  the  molecule  in  the  angle  of  the 
parallelopipedon  at  the  opposite  end  of  the  side  d  x,  at  the  time  t,  is 

du 

u  -\ — J—  'd x; 

dx 

and   hence   the  relative   velocity   of  the   two  molecules  is 


du     .  du     . 

u  -\-  -y—  •  dx  —  «  =  - —  a  X. 
dx  d  X 


At   the   time   t,  the   length   of    the   edge    joining  these   molecules   is 
dx,  and   at   the   end   of  the  time  t  +  d  t,  this  length  will   be 

du     ^        ,  7     /  ,     ,    '^  ^*     7  ,\ 

dx  -\ — ; —  •  dx  .  d  t  =  rt  ,r  (  1  +  -;—  •  d  t) : 
dx  ^  dx         ' 

the    second   term    being    the    distance    by    which    the    molecules    in 

question    approach    toward    or     recede     from     one     another    in    the 

time  dt. 

18 


274r  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

In  the  same  way  the  edges    of  the   parallelopipedon  which  at  the 
time  t,  were  dy  and  c?2,  become  respectively, 

d  V      ,       ,  1     /■,    ,    dv      .  . 

dy  +  -^'dy.dt     =dy{l+~'dt); 

dzo      ,        -  T     ,1     .     dw      .  . 

dz  +  —r-- dz.dt     =  dz  11  -{-  -j- -  dt)-, 
dz  ^  dz 

and  the   volume   of  the  parallelopipedon,  which   at   the   time   t,   was 
dx  .dy  .dz,  becomes  at   the   time  t  -^  dt, 

The   density,  which   was   D,  at   the    time  i,  being  a  function  oixyz 
and  t,  becomes   at   the   time  t  -\-  dt, 

^       dD     ,         clD     ^  dD     .         dD     . 

dt  dx  dy  dz 

which  may   be   put   under   the   form, 

/dD         dD   dx         dD    dy     ,    dD    dz\    . 
^  \dt    ^    dx     dt^    dy     dt    ^   dz     dt/       ' 

and  replacing 

dx        dy         dz 
dt         dt         d  t 

by  their  values  t^,  v,  w,  respectively, 

/dD         dD         ,   dD  clD     \   -, 

\  dt  dx  dy  dz       y 

Multiplying  this  by  the  volume  above,  we  have  for  the  mass  of   the 
parallelopipedon,  which  was 

D  .dx  .dy  .  dz, 

at  the   time   t,  the  value, 


fx.        ^dD        dD  dD         ,  dD       \    -   n 

L  \dt  d  X  dy  dz        /        J 

d  V 
dy 


x....y...(i+^.c.O-0  +  l;-'^')-0+'^-^'.) 


at  the  time   t  -\-  d  t. 


MECHANICS     OF    FLUIDS.  275 

But  these  masses  must  be  equal,  since  the  quantity  of  matter 
is  unchanged.  Equating  them,  striking  out  the  common  factors,  per- 
forming the  multiplication,  and  neglecting  the  second  powers  of  the 
differentials,  we  have 

-,,  { du    ,    dv        clw\       dD       dD  dD  dD 

\dx         dy        dz/         dt         dx  dy  d  z  ^       ' 

This  is  called  the  Equation  of  continuity  of  the  fluid.  It  expres- 
ses the  relation  between  the  velocity  of  the  molecules  and  the  den- 
sity of  the  fluid,  which  are  necessarily  dependent  upon  each  other. 
This  is  a  fourth  equation. 

§250.— If  the  fluid  be  compressible,  then  will  the  fifth  equation 
be   given   by  the   relation, 

F{D,p)  =  0, (402) 

as  is  illustrated  in  the  particular  instance  of  Mariotte's  law.  Equa- 
tion (389).  The  form  of  the  function  designated  by  the  letter  i^, 
will    depend    upon    the   nature    of  the   fluid. 

§251. — If  the  fluid  be  incompressible,  the  total  diflfcrential  of  Z> 
will   be   zero,  and 

dD     ,     dD  dD  dD 

and  consequently,  the  equation  of  continuity,  Equation  (401),  becomes, 

d  u     ,     dv  diti 

-7-+:!—   +-T-=Oj ('lO-l) 

dx  dy  dz  ^       ' 

St 

and   we   have   for  .the   determination    of  n,  v,  w,  D    and  jh    the   five 

Equations  (400),  (403),  (404). 

§  252. — These  equations  admit  of  great  simplification  in  the  case 
of  an  incomjjrcssible  homogeneous  fluid  when  u  •  dx -\-  v  .dy  -\-  iv  .d  z, 
is  a  perfect  differential.     For  if  we  make 

udx  -f  vdy  -\-  wdz  =  c?9. 


276  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

then   from   the   partial  differentials  will 

dS)  d  (p  d(D  /An-\ 

u  =  -j^;     v  =  -^;     w  =  -—-',    ....     (40o) 
ax  ay  az 

which,  in  Equation  (404),  gives  for   the  equation   of  continuity, 

'i!l  +  f?  +  !?;^  =  0; (m) 

dx"  dy  dz'^ 

by   the   integration  of  which  the   function  (p  may   be   found. 
Differentiating  the  values  of  ?«,  v  and  w  above,  we  have 

du  =  -r-^  ■    dv  =  —z— ;    dw  =  -J—  • 
d  X  dy  dz 

Eliminating  n,  v,  tv,  d  u,  d  v  and  d  w,  from  Equation  (400),  by  means 
of  the   values  of  these  quantities  above,  we  have 

1      dj)  d'^cp  dcp    d"cp         d(p    __^^^_g_  _  ^f_  ^     ^"^  ^ 

U'lTx    ~        ~  dx-dt  ~  ~d7"dx'^  ~  dy  '  dx.dy        dz     dx.dz' 

1      d}^  cP(p  dcr,        d'^cp  d(p    d-  o        d(p  ^     d^cp 

'D'~dy^  ~  dy.dt  ~  ~dy'  dy  .dx  d  y   d  y"^        dz     dy.dz' 

1      dp  d'^;p  dtp        d"(p  d^       d- (p dcp    d'^  cp  ^ 

'B'~d7~  ~   d  z  .dt~  ~dx'  dz  .dx  dy    dz.dy        dz     dz^ 

Multiplying  the  first  by  dx,  the  second  by  d y,  the  third  by  dz,  ad- 
ding and  reducing  by  the  relation  in  Equation  (40G),  we  find 

From  which,  by  integration,  may  be  found  the  pressure  at  any  point 
of  an  incompressible  fluid  mass  in  motion,  when  Equation  (4^6)  is 
the    equation  of  continuity.  • 

§253. — When  the  excursions  of  the  molecules  are  small,  the 
second  powers  of  the  velocities  may  be  neglected,  which  will  reduce 
Equation  (407)  to 

i- .  dp  =  Xdx  +  Ydy  -{-  Zdz  -d'^'     '     .     (408) 


MECHANICS     OF     FLUIDS.  277 

g254. — If  the  condition  expressed  by  Equation  (406)  be  not  ful- 
filled, then  Ave  must  have  recourse  to  Equation  (404)  to  find  the 
pressure. 

§255. — Resuming  Equation  (401),  which  appertains  to  a  compres- 
sible fluid,  retaining   the    condition  that 

udx  -f  vdy  +  IV  dz  =  da^ 

IS    a   perfect  differential,  and  from  which,  therefore, 

u  =  -y^;     V  =  -^•,     w=  -J-;      •      •     •     (409) 
dx  dy  dz 

we   obtain  by  substitution, 

(  fZw  dv        dio  I       dP      dP  d(?        dl)d(f>       dD'dcp  _ 

I   dx  dy        dz    )         dt        dx    dx         dy    dy         dz    dz 

If  the  excursions  of  the  molecules  from  their  places  of  rest  be 
very  small,  both  the  change  of  density  and  velocity  will  be  so 
small  that  the  products  which  constitute  the  last  three  terms  of 
this  equation  may  be  neglected,  and  the  equation  of  continuity  be- 
comes 

/  dii  d  V  ^^"\i'^-^^(\ 

^dx  dy  dz  '  d  t  ' 

and  replacing  du,  dv  and  dw,  by  their  values  from  Equations  (409), 
and  dividing  by  P,  we  find 

Uf^  +  !^  +  ^  +  !^=0.    .    .    .     (410) 
d  t  d x-         dy  dz- 

from  which,  together  with  the  equation  connecting  the  extraneous 
forces  with  the  co-ordinates  xyz,  and  that  expressive  of  Mariotte's 
law,  the  function  9  may  be  found,  then  the  value  of  P,  and  finally 
that  of  p. 

The  excursions  being  small,  if  we  impose  the  additional  condi- 
tion that   the   molecules  of  the   fluid   are   not   acted   upon   by   extra. 


278         ELEMENTS     OF     ANALYTICAL     MECHANICS. 

neous  forces,  in  which   case   the   motions   can   only  arise   from    some 
arbitrary  initial  disturbance ;    then,  Equation  (408), 

1  ^     ^"P   _        ^'P 

'l)'^~~      '  IT  ~  ~  ~dT' 

and   by  ]\Iariotte's   law, 

p  =  P.D  =  a^.D (411) 

in  which 


whence,  by  division, 


cnog^^_^ (413) 


which   substituted   above,  gives 

^=-(S  +  '5^  +  '^)-  •  •  •  ("^) 

From  this  Equation  the  function  9  is  to  be  determined,  then  the 
value  of  i>,  from  Equation  (410),  and  that  of  p,  from  either  of  the 
Equations  (411)  or   (413). 

§256. — If  the  fluid  be  confined  in  a  narrow  tube,  so  that  the 
motion  can  only  take  place  in  the  direction  of  its  axis,  the  co- 
ordinate axis  X  may  be  assumed  to  coincide  with  this  line;  in  which 
case  V  and  to  will   each  be  zero,  and.  Equation  (409), 

whence  Equation  (414)  becomes 

^  =  a-.^ (415) 

dt^  dx^  ^       ' 

To   integrate  this,  add  to   both   members 

rf2(p 
a  • r-? 

dx  •  d  t 


MECHANICS     OF     FLUIDS.  279 

and   we   shall   have 

dt        \dtr  dxJ        dx        \dt  dxJ  ' 

»#» 
and   making 

dt  dx 

we   have 

dV  dV 

d  t  a  X 

and   V  being   a  function  of  x  and  t,  we  have,  by  differentiating, 

dV      ,  dV     , 

dV=----dt  +  -j—-dx', 
at  dx 

dV  ■ 

or  by  substituting  for    — —   its  value   above, 

dV  =  ^  (dx  +  adt)  =  ^-dix^  at), 
d  X  at 

» 

and  by  integration, 

V=^  +  a-^=r{x+at), 
dt  dx 

m  which  F'  denotes    any   arbitrary  function. 
In  like    manner,  by  subtracting 

a  •  -. !—» 

d  t '  d  X 

from    both  members  of  Equation  (-llo),  we  find 

—r-  —  a  •  -J—  =  /  {X  —  a  t), 
dt  dx 

in  which  /'  denotes  any  arbitrary  function. 
Whence,  by  addition, 

-^  =  ^F'  {x  +at)  +  ^f  [x  -  at), 


280  ELEMENTS     OF    ANALYTICAL     MECHANICS. 

i 


and  by  subtraction, 


But 


^^2-.ri.  +  a,)-lf.i. 


d(D      , — ,     d(i)      , 

dca  =^  —r—  '  at  -\ ; ax 

^         dt  dx 


at). 


whence, 

(/^  _ F  {x+  at)d{x  +  at)  —  ——  -  f  {x  —  a  t)  d{x  —  at) 

ia  " 


and  by  integration, 


2a 


=  F{x  ^  at)  +f{x  -at) 


(416) 


in   which  F  and  /,  denote   any   arbitrary   functions  whatever,  and  are 
determined   from   the   initial    conditions   of  the   question. 

This  last  formula  is  used  in  discussing  the  subject  of  sound,  and 
the  more  general  equations  which  go  before  are  employed  in  devel- 
oping the  principles  of  light  and  heat  as  well  as  those  of  the  tidal 
waves  of  the  ocean   and   of  the  atmosphere. 


EQUILIBKIUM    OF    FLUmS. 


257. — If  the   fluid  be   at   rest,  then  will 

^-n.  ^-n-  ^-n. 
dt^-  -     '    df'  -     '    di^  -     ' 


and   Equations  (399)  become 

dp 
d  X 

dp 
dy 

dp 


=  D.X- 
=  D.Y; 
=  D.Z. 


(417) 


§258. — Multiplying  the  first  by  c?.r,  the  second   by  fZ  y,  the  third 
by   dz^  and  adding  we  find, 

dp  ^  D{Xdx -^  Ydy  +  Zdz);  '     •     •     •    (418) 


MECHANICS    OF    FLUIDS.  281 

and   by  integration, 

p  =  fn  .{Xdx  +  Ydy  ^  Zdz);    '     .     .     -(419) 

whence,  in  order  that  the  value  of  p  may  be  possible  for  any 
point  of  the  fluid  mass,  the  product  of  the  density  by  the  function 
Xdx  -f-  Ydy  +  Zdz^  must  be  an  exact  differential  of  a  function  of 
the  three  independent  variables  ar,  y,  z.  Reciprocally,  when  this  coudi- 
tion  is  fulfilled,  not  only  will  the  pressure  at  any  point  become  known 
by  substituting  its  co-ordinates,  but  the  Equations,  (417),  will  be  sat- 
isfied, and  the  fluid  will  be  in  equilibrio. 

g  259. — Conceiving  those  points  of  the  fluid  which  experience  equal 
pressures  to  be  connected  by,  indeed  to  form  a  surface,  then  in 
passing  from  one  point  to  another  of  this  surface,  we  shall  have 
dp  =  0,  and 

Xdx  +  Ydy  +  Zdz  =  0, (420; 

which   is  obviously   the   differential  equation   of  the   surface. 

Dividing  this  by  Eds,  in  which  E,  denotes  the  resultant  of  the 
forces  which  act  upon  any  particle,  and  ds,  the  element  of  any 
curve  upon  the   surface  passing  through   the   particle,  we  have 

M'  ds    '^  M'  ds    '^  M     ds  '  ^  ^  ' 

whence  the  resultant  of  the  forces  acting  upon  any  one  of  the 
elements  of  a  surface  of  equal  pressure,  is  normal  to  that  surface. 
This  is  the  characteristic  of  what  is  called  a  level  surface,  which 
may  be  defined  to  be  any  surface  which  cuts  at  right  angles  the 
directions  of  the  resultant  of  the  forces  which  act  upon  its  particles. 

g260. — If  Equation  (420)  be   integrated,  we   have 

\xdx  +  Ydy  +  Zdz)  =  C, (422) 


/( 


in  which  C  is  the  constant  of  integration.  The  niagnitudes  of  this 
constant  must  result  from  the  dimensions  of  the  surface,  or  from 
the   volume   of    the    ffuid    it    envelops.       By   giving    it    diffferent   and 


282  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

suitable  values,  we  may  start  from  a  single  particle  and  proceed  out- 
wards to  the  boundary  of  the  fluid,  and  if  the  successive  values 
differ  by  a  small  quantity,  we  shall  have  a  series  of  level  concentric 
strata. 

The  last  value  assigned  to  C  must  belong  to  the  bounding  sur- 
face, which  is  also  a  surface  of  equal  pressure ;  otherwise  the  co- 
ordinates of  this  surface  could  not  satisfy  Equation  (420),  and  con- 
sequently, Equations  (417)  and  (421),  and  the  surface  particles  could 
not  be  in  equilibrio,  which  would  be  contrary  to  the  hypothesis. 
Every  free  surface  of  a  fluid  in  equilibrio  is,  therefore,  a  level  sur- 
face. 

§261. — Putting  Equation  (418)   under   the  form 

^  =  Xdx  +  Ydy  -f  Zdz, (423) 

we  see-  that  whenever  the  second  member  is  an  exact  differential, 
p  must  be  a  function  of  i>,  since  the  first  member  must  also  be  an 
exact   differential.     Making,  therefore, 

P  =  F{Dl (424) 

in  which  F  denotes  any  function  whatever,  the  above  equation  be- 
comes 

z  Xdx  +  Ydy -^  Zdz;      •     •     •     (425) 

but  for  a  level  surface  or  stratum,  the  second  member  reduces  to 
zero  ;  whence, 

dF{D)  =  0; 

and    oy    integration, 

F{D)  =  C; 

whence,  not  only  will  each  level  stratum  be  subjected  to  an  equal 
pressure  over  its  entire  surface,  but  it  will  also  have  the  same 
density  throughout. 

§262. — If  the  fluid  be  homogeneous  and  of  the  same  temperature 
throufrhout,  then  will  D  be  constant,  and  the  condition  of  equilibrium 


MECHANICS    OF    FLUIDS.  283 

simply  requires  that  the  function  Xdx  +  Ydy  -(-  Zdz,  Equation 
(419),  shall  be  an  exact  dificrential  of  the  three  independent 
variables  x,  ?/,  z,  and  -when  this  is  not  the  case,  the  equilibrium 
will  be  impossible,  no  matter  ■what  the  shape  of  the  fluid  mass, 
and    though  it  were   contained   in   a   closed   vessel. 

But  the  function  above  referred  to  is,  §  133,  always  an  exact 
differential  for  the  forces  of  nature,  which  are  either  attractions  or 
repulsions,  whose  intensities  are  functions  of  the  distances  from  the 
centres  through  which  they  are  exerted.  And  to  insure  the  equi- 
librium, it  will  only  be  necessary  to  give  the  exterior  surface  such 
shape  as  to  cut  perpendicularly  the  resultant  of  the  forces  which  act 
upon  the  surface  particles.  This  is  illustrated  in  the  simple  example 
of  a  tumbler  of  water,  or,  on  a  larger  scale,  by  ponds  and  lakes 
which  only  come  to  rest  when  their  upper  surfaces  are  normal  to 
the  resultant  of  the  force  of  gravity  and  the  centrifugal  force  arising 
from   the   earth's  rotation   on   its  axis. 

In  the  case  of  a  heterogeneous  fluid  subjected  to  the  action  of  a 
central  force,  its  equilibrium  requires  that  it  be  arranged  in  concentric 
level  strata,  each  stratum  having  the  same  density  throughout.  And 
the  equilibrium  will  be  stable  when  the  centre  of  gravity  of  the 
whole  is  the  lowest  possible,  §  134,  and  hence  the  denser  strata  should 
be   the   lowest. 

When  the  fluid  is  incompressible,  the  density  may  be  any  function 
whatever  of  the  co-ordinates  of  place.  It  may  be  continuous  or  dis- 
continuous. "When  it  is  given,  the  value  of  the  pressure,  is  found  from 
Equation  (419). 

§  263. — In   compressible   fluids   the   density   and  pressure  are   con- 
nected  by    law,  and   the   former  is  no  longer  arbitrary. 
Dividing  Equation  (418)  by  Equation  (389),  we  have 

d})  Xdx  -\-   Ydy  -\-  Zdz 


Integrating, 


rXdx -\-  y  dy  +  Zdz 
\ogp  =  J p         +log(7;.     .    .(42G) 


2Si         ELEMENTS     OF     ANALYTICAL    MECHANICS, 
denoting  the  base  of  the  Naperian  system  by  e,  we  have 

fXdi+Ydy+Zdz  .  (io'y\ 

■p  =  C.e-'  P  ' v'*^'^ 

and    this  substituted  in  Equation  (389),  gives 

rXdx  +  Ydy  +  Zdz 

D  =  ^— -, (428) 

These  equations  determine  the  pressure  and  density. 

For  any  surface  of  constant  pressure,  the  exponent  of  e,  in  Equa- 
tion (427),  must  be  constant,  its  differential  must,  therefore,  be  zero, 
and  all  the  consequences  deduced  from  Equation  (420)  will  follow ; 
that  is,  when  the  fluid  is  at  rest,  it  must  be  arranged  in  level  strata, 
each  stratum  having  the  same  density  throughout,  with  the  addition 
that  the  law  of  the  varying  density  must  be  continuous  by  the  re- 
quirements of  Mariotte's  law. 

If  the  temperature  vary,  then  will  P  vary,  and  in  order  that 
Equation  (427)  may  be  an  exact  differential,  P  must  be  a  function 
of  xrj  z,  and  hence.  Equations  (427)  and  (428),  when  p  is  constant, 
D  will  be  constant;  that  is,  each  level  stratum  must  be  of  uniform 
temperature  throughout. 

It  is  obvious  that  the  atmosphere  can  never  be  in  equilibrio  ;  for 
the  sun  heating  unequally  its  different  portions  as  the  earth  turns 
upon  its  axis,  the  layers  of  equal  pressure,  density  and  temperature 
can  never  coincide.  Hence,  those  perpetual  currents  of  air  known  as 
the  trade  ivmds,  and  the  periodical  monsoons ;  also,  the  sea  and  land 
breezes,  variable  winds,  &c.,    &c. 

§  264. — Rest  is  a  relative  term ;  when  applied  to  a  particle  of  a 
fluid  mass,  it  means  that  that  particle  preserves  unaltered  its  place  in 
regard  to  the  other  particles;  a  condition  consistent  with  a  bodily 
movement  of  the  entire  mass. 

If  a  liqm'd  mass  turn  uniformly  about  an  axis,  the  preceding 
equations  will  make  known  its  permanent  figure.  For  this  purpose 
it  will  be  sufficient  to  join  to  the  forces  X,  Y,  Z,  the  centrifugal  force. 


MECHANICS     OF     FLUIDS. 


285 


Take   the  axis  z  as  the  axis  of  rotation ;    denote  the  angular  velocity 
by  (p,  and    the  distance  of 
the   particle    M   from    the 
axis  z  by  r;  then  will 

r2  =  0:2  ^  yi . 

the  centrifugal  force  of  M 
regarded  as  a  unit  of  mass, 
will  be 

r(p2, 


and  its  components  in  the  -^ 

direction  of  x  and  y,  respectively, 


r 


7* .  (p**  •  —  ■=.  y  (^^ 


and  these  in    Equation  (418),  give 

dp  =  D.{Xdx  +  Ydy  +  Zdz  +  (f.xdx  +  92  y .  (fy). .  (409) 

When  the  second  member  is  an  exact  differential,  the  permanent  form 
Avill    be   possible. 

For    the    free   surface  dp  =  0,  and  we  have 

Xdx  +  Ydy  +  Zdz  +  (^"".x.dx  +  '^''ydy  =  0-  •  -(430) 
Example  1.— Let   it  be  required     to    find    the   figure    assumed   by 
the  free  surface    of  a    heavy  and   homogeneous  fluid    contained  in    an 
open  vessel  and  rotating  about   a  vertical    axis. 
Here, 

X=  0;     r=  0;     Z=  -  g; 

and    Equation  (430)  becomes 

gdz  =  <p2(xrf.r  +  ydy). 
Integrating, 


=  |^(-^^  +  y^-)+  <^'5 


(431) 


which  is  the  equation  of  a  paraboloid  wliosc  axis  is  that  of  rotation. 


286 


ELEMENTS    OF    ANALTTICAX    MECHANICS. 


To  find  the  constant  C,  let  the  vessel  be  a  right  cylinder,  with 
circular  base,  whose  radius  is  a,  and  denote  by  h  the  height  due  to 
the   velocity  of  the  fluid  at  the  circumference,  then 


and 


a2(p2  _  2gh, 


s  =  -^  +  C 
a- 


(432) 


Denote  by  b  the  height  of  the  liquid  before  the  rotation ;  its 
volume  will  be  rt  a?  .  b.  Conceive 
the  whole  body  of  the  liquid  to 
be  divided  into  concentric  cylin- 
drical layers,  having  for  a  common 
axis  the  axis  of  rotation.  The  base 
of  any  one  of  these  layers  will 
have  for  its  area,  neglecting  dr^, 
2'7fr.  dr,  and  for  its  volume,  taking 
the  origin  of  co-ordinates  in  the 
bottom  of  the  vessel,  2ifr.dr.z, 
which  being  integrated  between  the 
limits  r  =  0  and  r  =  a,  will  give 
the  whole  volume  of  the  fluid,  and 
hence. 


xr 


a26  =  ^fzr.dr  +  C ; 


replacing  r .  d  r  \)\  its  value  from  Equation  (432),  and  integrating 
between  the  limits  z  =i  C  and  2  =  A  -f  C,  which  are  the  values 
given    by  Equation  (432)  for  r  =  0    and   r  =  a,  we  find 

C=b-lh, 

and   the   equation  of  the   upper   surface  becomes 

A  /-^         ,  , 

a''  ■* 

The   least  and  greatest   values   for   z,  are   b  —  \h    and    b  -{-  ^h, 
obtained  by  making  r  =  d  and  r  ==  a,  so  that  the  depression  of  the 


MECHANICS    OF    FLUIDS. 


287 


liquid  at  the  axis  is  equal  to  its  elevation  at  the  surface  of  the 
cylindrical  vessel,  and  is  equal  to  half  the  height  due  to  the 
velocity  of  the  latter. 

§  ^Q,b.— Example  2.— Let 
the  fluid  elements  be  attract- 
ed to  the  centre  of  the  mass 
by  a  force  varying  inversely 
as  the  square  of  the  distance. 
Talce  the  origin  at  the  cen- 
tre ;  denote  the  distance  to 
the  particle  m  from  that  point 
by  r,  and  the  intensity  of  the 
attractive  force  at  the  unit's 
distance  by  k.     Then  will 

k                               X 
P  z=  m  -r  ;      cos  a  ;= 


cos  j3  = —  ;     cos  y  =  — 


and 


Jcx. 


ky 


X  = r  ;     Y  —  —  -^  ;     Z  —        —  ; 


which  in  Equation  (430),  give 

—  {xdx  -\-  tjdy  +  zdz)  —  (f  {x  d  x  +  ydy)  =  0, 


^Ji^^^U'- +  ,/.)  =  <,, 


and  by  integration, 


making 


r  z 


a;2  -I-  y2  _  f.2  cos2  6^ 

in  which  ^'denotes   the  angle  made  by  r,  with   the  plane  x y, 

V      '      9. 


288  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and    denoting    the   distance   from   the   origin    to   the   point   in   which 
the  free  surface  cuts  the  axis  z  by  unity,  we  have,  by  making  (5  =  90°, 

1  -  ^' 

•which   substituted   above,  and   solving   with  respect   to  cos^^,  gives 

^<p2.cos2  0  :=ifc-^^ (434) 

and   making  r  =  1  +  «,  we  have 

Tcu 


\  (p2  •  cos^  ^  ■=. 


(1  +  uf 

If  the  angular  velocity  be  small,  then  will  u  be  very  small. 
Developing  the  second  member,  with  this  supposition,  and  limiting 
the    terms   to    the  first   power  of  «,  we  find 

J  92  .  cos2  (3  =  A:  (tt  -  3  w2). (434)' 

xVeglecting  3  m^,  and  replacing  xi  by  its  value,  viz.:  r  —  1,  we 
have   for  a   first   approximation, 

(p2 
?•    =    1    +    Vt   ■  C0S2  d. 

2/fc 
From  Equation  (434)',  we  find 

2)2  .'cos^r 


+  3m2, 


~        'Ik 
and   this   in   the   equation 

r  —  \  ^r  V, 
gives 

r  =  1  +  |v  •  cos2  ^  +  3  «2  ; 

,       (p*  •  cos"*  ^       ^  , 

and  replacing  v?  by  its  approximate  value  — —7 )    above,  by  neg- 
lecting 3  ^2,  we   have 

(p2  3  (p4  .  cos*  ^ 

for    the    polar  equation  of  the   meridian  section. 


MECIIAXIC3     OF    FLUIDS. 


289 


Comparing    this    with   the    equation 


1  +  I  e2  cos2  Q  ^  s.  e*  .  cos*  t)  +  &;c,, 


they   become  identical    by    neglecting    the  higher  powers    and  making 


The  free  surface  of  the  fluid  approximates  therefore  very  closely 
to  an  ellipsoid  of  revolution  of  which  the  eccentricity  of  its  meridian 
section  is  equal  to  the  square  root  of  the  quotient  arising  from 
dividing  the  centrifugal  force  at  the  unit's  distance  from  the  axis 
of  rotation,  by  the  force  of  attraction  at  an  equal  distance  from  the 
centre. 


PRESSURE   OF   HEAVY   FLUIDS. 

go(5(5_ — When  a   fluid   contained  in   any   vessel  is   acted   upon   by 
its  own  weight,  if    the    axis  z  be    taken   vertical 
and  positive  downwards,   then  will 

and    Equation  (418)  becomes,  after  integrating, 

p=Dffz-i-  C; 

f 

and    assuming    the    plane   .ry    to    coincide    with 

the   upper    surfccc    of  the    fluid,  which    must,    when    In    equilibrio,   be 

horizontal,  we   have,  by  making  z  =  0, 

in  which  p'  denotes  the  pressure  exerted  upon  the  unit  of  the  free 
surface.     Whence, 

p~p^  =  D.ff.z. (435) 

The  first  meml)er  is  the  pressure  exerted  iq>on  a  unit  of  surface, 
every  point  of  which  unit  having  a  pressure  equal  to  that  sustained 
bv   the    element   whose    co-ordinate    is   z. 

19 


290 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


If  p'  =  0,  then    will 

P  =  DU^~\ (436) 

and  denoting  by  b  the  area  of  the  surface  pressed,  and  by  db,  the 
element  of  this  surface,  whose  co-ordinate  is  z,  we  have,  Equation 
(39T),  for  the   pressure   upon   this   element   denoted   by  p^, 

T,  =  Bg.z.db, 

and  the  same  for  any  other  element  of  the  surface ;  whence,  deno- 
ting   the    entire   pressure  by  P,    we    shall   have 

P  ^^.p,  ^  Dcj.l^z.db. (437) 

But  if  Zi  denote  the  co-ordinate  of  the  centre  of  gravity  of  the 
entire  surface  6,  then    will,  Equations  (91), 


and 


db  =  bz,, 


P  ^  Dg.b, 


(438) 


Now  b  z,  is  the  volume  of  a  right  cylinder  or  prism,  Avhose  base 
is  6,  and  altitude  2^;  Dg.b.z^  is  the  weight  of  this  volume  of 
the  pressing  fluid.  Whence  we  conclude,  that  the  inesmre  exerted 
vpon  any  surface  by  a  heavy  jiu'id  is  equal  to  the  weight  of  a.  cylin- 
drical or  prismatic  cohfmn  of  the  fluid  whose  base  is  equal  to  the 
surface  pressed,  and  u'hose  altitude  is  equal  to  the  distance  of  the  cen- 
tre   of  gravity    of  the    surface  beloio  the   vjyj^er   surface    of  the  fluid. 

When  the  surface  pressed  is  horizontal,  its  centre  of  gravity  will 
be  at  a  distance  fi'om  the  upper  surface  equal  to  the  depth  of  the 
fluid. 

This  result  is  wholly  independent  of  the  quantity  of  the  pressing 
fluid,  and  depends  solely  upon  the  density  of  the  fluid,  its  height,  and 
the    extent   of  the    surface   pressed. 

Example  1.  —  Required  the  pressure 
against  the  inner  surfixce  of  a  cubical  ves- 
sel filled  with  water,  one  of  its  flices  being 
horizontal.  Call  the  edge  of  the  cube  a, 
the  area  of  each  face  will  be  o^,  the  dis- 
tance of  the  centre  of  gravity  of  each 
vertical     face    below    the    upper    surface  will    be   \a,    and    that    of  the 


l\ 

'■■■\ 

MECHANICS    OF    FLUIDS. 


291 


lower    fixcc   a ;      -whence,    the    principle    of    the     centre    of    gravity 
gives, 


5a2 


3 

5" 


Again, 


h  :=  5  a^  ; 

and   these,  substituted  in  Equation  (438),  give 

P  =  D  .g-h.z,  =  D.g.^a\ 

Now  Dg  X  P  =  2>^,  is  the  weight  of  a  cubic  foot  of  water  =  62,5 
lbs.,  whence, 

lbs. 

P  =  62,5  X  3a3. 
Make   a  —  1  feet,  then  will 

lbs. 

P  =  62,5  X  3  X  (T)3  =  27562,5. 

The  weight  of  the  water  in  the  vessel  is  62,5  a^,  yet  the  pressure 
is  62,5  X  3a3,  whence  we  see  that  the  outward  pressure  to  break 
the  vessel,  is   three  times    the   weight   of  the    fluid. 

Example  2. — Let  the  vessel  be  a  sphere  filled  with  mercury,  and 
let  its  radius  be  R.  Its  centre  of  gravity  is 
at  the  centre,  and  therefore  below  the  upper 
surface  at  the  distance  R.  The  suriace  of  the 
sphere  being  equal  to  that  of  four  of  its 
great  circles,  we  have 

6  =  4*i22; 


whence, 

and.  Equation  (438), 


b.z,  =  4*i22j 


P  ^Art  .D.g.R^. 

The  quantity  Dg  x  P  =  i>^,  is  the  weight  of  a  cubic  foot  of 
mercury  =  843,75  lbs.,  and  therefore,  substituting  the  value  of 
ff  =  3,1416, 

lbs. 

P  =  4  X  3,1416  X  843,75  .  R\ 


292 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


Now  suppose  the  radius  of  the  sphere  to  be  two  feet,  then  will 
i23  —  8,  and 

lbs.  lbs. 

P  =  4  X  3,141G  -  843,75  X  8  =:  84822,4. 

The  volume  of  the  sphere  is  f  *  i^^ ;  and  the  weight  of  the  con- 
tained mercury  will  therefore  be  ^ir  E?  g  D  —  W.  Dividing  the 
whole   pressure   by  this,  we  find 

P 


W 


=  3 


whence   the   outward    pressure   is  three  times  the  weight  of  the  fluid. 

Exami~)U  3. — Let   the  vessel  be   a    cylinder,  of    which    the    radius 
r  of  the  base   is   2,    and    altitude    I,   G    feet.     Then    will 

h.z,^  ':rrl{r  +  I)  =  3,1410  X  2  X  G  X  8; 

which,  substituted  iu  Equation  (43S), 

F  =  301,593G  X  Dg, 
and 

W  =  3,1410  X  22  X  0  X  i)^  =  75,398  X  Dff', 
whence, 

F    __  301,5930  X  F(f  _ 
W  ~     75,3964.  X»^     ""      ' 

that   is,  the  pressure    against  this    particular  vessel    is  four    times    the 
weight   of  the    fluid. 

g  267.— The  point  through  which  the  resultant  of  the  pressure 
upon  all  the  elements  of  the  surface 
passes,  is  called  the  centre  of  pressure. 
Let  EIF  be  any  plane,  and  MN 
the  intersection  of  this  plane  produced 
with  the  upper  surfoce  of  the  fluid 
which  presses  against  it.  Denote  the 
area  of  any  elementary  portion  n  of 
the  plane  EIF  by  c?6  ;  and  let  m  be 
the   projection   of    its   place   upon   the  -^ 

upper  surface  of  the  fluid ;  draw  m  M 
perpendicular  to  M N^  and  join  n  with  Mhj   the  right  line  n  M,  the 


JJ^ 


JV 


PA 


MECHANICS    OF    FLUIDS.  293 

latter  -will  also  be  perpendicular  to  M N,  and  the  angle  n  M  m  -will 
measure  the  inclination  of  the  plane  E IF  to  the  surface  of  the 
fluid.  Denote  this  angle  by  (p,  the  distance  mn  by  //,  and  Ma  by  r' ; 
then   Avill 

h'  =1  r'  sin  (p  ; 

the   pressure    upon    the    element   d  /v, 

D  (J  .r'  sin  9  dh\ 

its    moment   with    reference   to    the    line  M N^ 
D  gr"^  sin  <p  ,  (Z6  ; 

and  for   the   entire  surface,  the   moment   becomes 
2?^,  sin  9  .  2  r'-  rf6. 

Denote  by  r  the  distance  of  the  centre  of  gravity  of  the  surface 
pressed,  from  the  line  M  N,  its  distance  below  the  upper  surface  of 
the  fluid  -will  be  r .  sin  cp ;  and  the  pressure  upon  this  surface  will  be 

D  g  .  r  sin  9  .  i  ; 

and  if  I  denote  the  distance  of  the  centre  of  pressure  from  the 
line  M N,  then  will 

Dg  .rs'm(p.b.l  =  Dg  .  sin  9  .  2  r^, 
from  which  we  have, 

1--=^^; (439) 

whence,  Equation  (2G4),  the  centre  of  pressure  is  found  at  the  centre 
of  pei'cussion  of  the   surface  pressed. 

§208. — The  principles  which  have  just  been  explained,  are  of 
great  practical  importance.  It  is  often  necessary  to  know  the  pre- 
cise amount  of  pressure  exerted  by  fluids  against  the  sides  of  ves- 
sels  and  obstacles  exposed  to  their  action,  to  enable  us  so  to  adjust 
the  dimensions  of  the  latter  as  to  give  them  suflicient  strength  to 
resist.  Reservoirs  in  which  considerable  quantities  of  water  are  col- 
lected and  retained  till  needed  for  purposes  of  irrigation,  the  supply 
of  cities   and    towns,  or   to    drive  machinery  ;    dykes  to  keep   the  sea 


294 


ELEMENTS     OF    AIv^ALYTICAL    MECHANICS. 


and  lakes  from  inundating  low  districts ;  artificial  embankments  con 
structed  along  the  shores  of  rivers  to  protect  the  adjacent  country 
in  times  of  freshets ;  boilers  in  which  elastic  vapors  are  pent  up  in 
a  high  state  of  tension  to  propel  boats  and  cars,  and  to  give  motion 
to    machinery,  are  examples. 

1 269. — As  a  single  instance,  let  it  be  required  to  find  the  thick- 
ness of  a  pipe  of  any  material  necessary  to  resist  a  given  pres- 
sure. 

Let  ABC  be  a  section  of  pipe  perpen- 
dicular to  the  axis,  the  inner  surface  of 
which  is  subjected  to  a  pressure  of  2^  pounds 
on  each  superficial  unit.  Denote  by  R  the 
radius  of  the  interior  circle,  and  by  I  the 
length  of  the  pipe  parallel  to  the  axis ; 
then  will  the  surface  pressed  be  measured 
by  2  *  i2 .  Z ;  and  the  whole  pressure  by 
2'g  R.l.p. 

By  virtue  of  the  pressure,  the  pipe  will  stretch ;  its  radius  will 
become  R  +  d  R,  the  path  described  by  the  pressure  will  be  d  R, 
and   its   quantity  of  work 

2*i2.  l.2)dR. 

The  interior  circumference  before  the  pressure  was  2'^R,  afterwards 
2Tr  [R  +  dR),  and  the  path  described  by  resistance,  2'^dR.  And 
if  B  denote  the  resistance  which  the  material  of  the  pipe  is  capable 
of  opposing,  to  a  stretching  force,  without  losing  its  elasticity  over 
each  unit  of  section,  t  the  thickness  of  the  pipe,  then,  by  the  prin- 
ciple of  the   transmission  of  work,  must 


whence. 


2'X.B.l.dR.t  =  2'gR.l.p.dR; 
Rp 


t  = 


B 


The   value  of  p   is   estimated   in   the   case   of  water   pressure   by 
the  rules  just  given.     That  in  the  case  of  steam  or  condensed  gases, 


MECHANICS     OF     FLUIDS, 


295 


bv  rules  to  be  given  presently.  The  value  of  B  is  readily  obtained 
from  Table  I,  giving  the  results  of  experiments  on  the  strength  of 
materials. 


EQUILLBKR-il   A>'D   STABILITY   OF  FLOATES'G   BODIES. 

8  270 — "When  a  body  is  immersed  in  a  fluid  it  is  not  only 
acted  upon  by  its  own  weight,  but  also  by  the  pressure  arising  from 
the  weight  of  the  fluid,  and  the  circumstances  of  its  rest  or  motion 
will  be    made    known    by   Equations  {A)  and  {B). 

Let  ED  be  the  body  ;  take  the  plane  x  y  in  the  plane  of  the  up- 
per surface  of  the  fluid, 
supposed  at  rest,  and 
the  axis  of  2  therefore 
vertical.  Denote  by 
h  the  entire  surface 
of  the  body,  and  by 
d  b,  one  of  its  elements, 
whose  co-ordinates  of 
position  are  x  y  z.  The 
pressure  upon  this  ele- 
ment will  be 

D.g  .z.db, 

in  which  I)  is  the  density  of  the  fluid,  and  g  the  force  of  gravity. 

This  pressure  is,  §  248,  normal  to  the  surface,  and  denoting  by 
a, /3  and  7,  the  angles  which  this  normal  makes  with  the  axes  x  y  z, 
respectively,  the  components  of  the  pressure  in  the  direction  of  these 
axes  will   be 

D-g  .z  .db  .cosa;     D  .  g  .  z  .d  b  .cos  [3  ;     D  .  g .  z .  d  b  .  cosy. 

Similar  expressions  being  found  for  the  components  of  the  pressure  o)\ 
other  elements,  we  have,  by  taking  their  sum, 

Dg.Ilz.db.  cos  a;     I)  g  .'E  z  .  db  .cos  (3  ;     D  g  ."S.  z  .  db  .  cosy. 

But  db.  cos  a,    f/i.cos/3,    and    db.  cosy,    are    the  projections    of  the 
area  db  on  the  co-ordinate   planes  z  y,  z  x  and  x  y,  respectively;    and 


296  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

2  z.db.  cos  a,  2  z.db. cos  [3^  Iz.db. cosy,  are  the  volumes  of  right  cylin- 
ders or  prisms,  whose  bases  are  the  projections  of  the  entire  surface 
pressed  upon  the  same  co-ordinate  planes,  and  whose  common  altitude 
is  equal  to  the  distance  of  the  centre  of  gravity  of  this  projection  from 
the  upper  surface    of  the   fluid. 

Whence  we  conclude,  ihut  the  component  of  ike  pressure  on  any 
surface,  estimated  iu  any  direction,  is  equal  to  the  pressure  on  so  much 
of  that  surface  as  is  equal  to  its  projection  on  a  plane  at  right  angles 
to    the   given   direction. 

The  cylinder  or  prism  which  projects  an  element  on  one  side  of 
the  body  will  also  project  an  element  situated  on  the  opposite  side  ; 
these  projections  will,  therefore,  be  equal  in  extent,  but  will  have 
contrary  signs,  for  the  normal  to  the  one  will  make  an  acute,  and 
to  the  other  an  obtuse  angle  with  the  axis  of  the  plane  of  projection. 
When  these  projections  are  made  upon  any  vertical  plane,  the  value 
of  z  will  be  the  same  in  both,  and  hence,  for  each  positive  product, 
z  .  db .  cos  a  and  z  .  db  .  cos  p,  there  will  be  an  equal  negative  product ; 
therefore, 

Dg  .IsZ  .db.cosa.=z'^Pcosa.  =  Q;  D  g  .'S  z  .db  .  cos  (3  —  1,  Pcos  ^  =  0. 

That  is,  the  sum  of  the  horizontal  pressures  in  the  directions  of 
X  and  y,  and  therefore  in  all  horizontal  directions,  will  be  zero  ;  and 
the   first   and   second   of  Equations  (120),  give 

d'^  X         „  d~  1/ 

or,  Avhich  is  the  same  thing,  there  can  be  no  horizontal  motion  of 
translation   from   the   fluid    pressure. 

When  the  projections  of  opposite  elements  are  made  upon  a 
horizontal  plaaie,  they  will  still  be  equal  with  contrary  signs,  the 
normal  to  the  elements  on  the  lower  side  making  obtuse,  while  the 
normals  to  the  elements  above  make  acute  angles  with  the  axis  z; 
but  the  corresponding  values  of  z  will  differ,  and  by  a  length  equal 
to  that  of  the  vertical  filament  of  the  body  of  which  these  element'; 
form   the   opposite   bases,  and   hence 

D g  .1,  z  .db  .  cosy  z=  D g .:^  {z'  —  z^)  d b  cosy  =  —  D gl  cd bcosy  ■  {-i^r) 


MECHANICS    OF    FLUIDS.  297 

in  \vhich  z'  denotos  the  ordinate  fur  the  ujiper,  and  z^  that  for  the 
lower  clement  in  the  same  vertical  line,  and  c  the  distance  between 
the  elements;  and    the    third  of  Equations  (120)  becomes 

2  (  P  cos  7  —  m  •  ——  )  =  Mg  —  D  g  •!  c  ■  d  b  •  cos  y  —  2  ??i  •  --  -  =  0. 

But  I.C .  d  Ij  .  cosy  is  the  volume  of  the  immersed  body  \vhich  is 
obviously  equal  to  that  of  the  displaced  fluid;  also  D  <j  .1c  .d  b  .cos,y 
is  the  weight  of  the  displaced  fluid ;  and  ^fg  that  cif  the  body. 
Denoting  the  volume  of  the  body  by  V\  its  density  by  i>',  the 
above   may  be  written 

F'i)'r/  -    V  Drj  _  2  m. -^1  =  0.      •     •     •     (441) 


Now.  when 


then  will 


V'D'g  -  V'Do  =  0, 


d-^z 

2  m  — --  =  0 

dfi 


and    there   can    be   no    vertical   motion    of  translation    from    the    fluid 
pressure  and   the  body's   weight. 
When  D'  >  D,  then  will 

d  fi  ' 

and    the   body  will    sink  with    an   accelerated    motion. 
When  D'  <  D,  then  %\  ill 

and    the    body  will  rise  Mith  an  accelerated  motion    till 


dC~ 


m--^^=V'D'g-VDg  =  0;     •     .     .     (442) 


298 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


in  which  V  denotes  the  volume  A  B  C,  of  the 
fluid  displaced.     At   this   instant  we  have 

V  D'cj  =z  VDc/;  .     .     .     (443) 

and  if  the  body  be  brought  to  rest,  it  will 
remain  so.  That  is,  the  body  will  float  at  the 
surface  when  the  weight  of  the  fluid  it  dis- 
places is  equal  to  its  own  weight. 

The  action  of  a  heavy  fluid  to  support  a  body  wholly  or  partly 
immersed  in  it,  is  called  the  buoyant  effort.  The  intensity  of  the 
buoyant  eflbrt  is  equal    to  the  weight  of  the  Jiaid  disjjlaced. 

Substituting  the  values  of  the  horizontal  and  vertical  components 
of  the   pressures  in  Equations  (118),  and  reducing  by  the  relations, 


Dff.'Ec.db.  cos  Y  .  x'  =  D  g  .  V .  x; 
D  (J  ."Lc  .  db  .  cos  7 . 2/'  =  D  g  .  V .  y  ; 


(444) 


in  which  x  and  y  are  the  co-ordinates  of  the  centre  of  gravity  of  the 
displaced  fluid  referred  to  the  centre  of  gravity  of  the  body,  we  find 


2  m 


2  m 


2  7n 


y'.d'^x'  -  x'  .d'-y' 


df 

z'-d-'x'  -  x' 

•  fZ2  z' 

df^ 

y' .  (?2  z'  _  z' 

.duy 

df^ 


=  0; 


(445) 


Equations  (444)  show  that  the  line  of  direction  of  the  buoyant 
effort  passes  through  the  centre  of  gravity  of  the  displaced  fluid. 
This  point  is  called  the  centre  of  buoyancy.  And  from  Equations 
(445),  we  see  that  as  long  as  x  and  y  are  not  zero,  there  will  bo 
an  angular  acceleration  about  the  centre  of  gravity.  At  the  instant 
X  —  ^  and  ?/  =  0,  that  is  to  say,  when  the  centres  of  gravity  of 
the  body  and  displaced  fluid  are  on  the  same  vertical  line,  this 
acceleration  will  cease,  and  if  the  body  were  brought  to  rest,  it 
would   have   no    tendency   to    rotate. 

To    recapitulate,  we  find, 


MECHANICS     OF     FLUIDS. 


299 


1st.  TJiat  the  ineasures  uiion  the  surface  of  a  hody  immersed  in 
a  heacy  fuid  have  a  sbi'jlc  resultant,  called  the  buoyant  effort  of  the 
fluid,  and   that    this  resultant   is   directed    vertically    upwards. 

2cl.  That  the  buoyant  effort  is  equal  in  intensity  to  the  weight  of 
the  find   dis2)laced. 

3d.  That  the  line  of  direction  of  the  buoyant  effort  jmsses  through 
the   centre    of  gravity  of  the  displaced  fuid. 

4th,    That  the   horizontal  j^rfs^^n res    destroy    one    another. 

§271. — Having  discussed  the  equilibrium,  consider  next  the  sta- 
bility of  a  floating  body.  The  density  of  the  body  may  be  homo- 
geneous or  heterogeneous. 
Let  AB  CD  be  a  section 
of  the  body  by  the  upper 
surface  of  the  fluid  Avhen 
the  body  is  at  rest,  G 
its  centre  of  gravity,  and 
H  that  of  the  fluid  dis- 
placed. Denote  by  V  the 
volume  of  the  displaced 
fluid,  and  by  M  the  mass 
of   the    entire    body.     The 

body  being   in  equilibrio,  the  line   G  JI  will  be   vertical,  and  denoting 
the    density  of  the    fluid   by  I),  mc    shall   have 


J/  =  i> .  T^ 


(44G) 


Suppose  the  section  A  B  CD  either  raised  above  or  depressed 
below  the  surface  of  the  fluid,  and  at  the  same  time  slightly  careened ; 
also  suppose,  Avhen  the  body  is  abandoned,  that  the  elements  have 
a  slight  velocity  denoted  by  u,  u',  iSjc.  Now  the  question  of  sta- 
bility will  consist  in  ascertaining  whi'thir  the  body  will  return  to  its 
former  position,  or  will    depart  more    and   more  from    it. 

The  free  surface  of  the  fluid  is  called  tlie  plane  of  floatation, 
and  during  the  motion  of  the  body  this  plane  will  cut  from  it  a 
variable   section. 

Let   A'  B'  C  D'  be  one  of  these  sections   at  anv  given   instant  of 


300  ELEMEISTS     OF    ANALYTICAL    MECHANICS. 

time  ;  A  B"  CD",  another  variaLle  section  of  the  body  by  a  hori- 
zontal  phme  through  the  centre  of  gravity  of  the  primitive  section 
ABCD,  and  A  C  the  intersection  of  the  two.  Denote  by  d  the 
inclination  of  these  two  sections,  and  by  ^  the  vertical  distance  of 
A  B"  C D'\  from  the  plane  of  floatation,  which  now  coincides  with 
A'  B'  C  D',  this  distance  being  regarded  as  negative  or  positive,  ac- 
cording as  A  B"  CD"  is  below  or  above  the  plane  of  floatation. 
The  variable  Cjuantities  6  and  ^  will  be  supposed  very  small  at  the 
instant  the  body  is  abandoned.  Will  they  continue  so  during  the 
whole  time  of  motion  1 

From  the  principles  of  livmg  force  and  quantity  of  work,  we  have, 
Equation  (121), 

y«2  .  d  M  -  2f{Xd  X  -\-  Ydy  +  Z  dz)  +  C. 

The  forces  acting  are  the  weights  of  the  elements  dM  and  the  verti- 
cal pressures,  the  horizontal  pressures  destroying  one  another ;  whence, 
A"  =0,     Y  =  0,    and 

fu^  dMzz:2  fzdz  +  C  =2^Zz  +  C.     .     •     (447) 

The  force  which  acts  upon  an  element  above  the  plane  of  floata- 
tion is  its  own  weight,  and  the  force  which  acts  upon  any  element 
below  that  plane  is  the  difference  between  its  own  weight  and  that 
of  the  fluid  it  displaces;  the  first  will  be  g .  d M^  and  the  second, 
g .  D  .d  V^  in  which  d  V  is  the  volume  of  d  M\    whence, 

I^Zz  =fg.z.dM-fgD.z.dV.      •     •     •     (448) 

But,  drawing  from  the  centre  of  gravity  O,  of  the  body,  the  perpen- 
dicular 0  E^  to  the  plane  of  floatation  A'  B'  C  D',  and  denoting  G  E 
by  2^,  we  have 

I  g  .z  .d  M  =^  g  Mz^. 

The  integral  I  g  D.z.d  F,  will  be  divided  into  two  parts,  viz:  one 
relating  to  the  volume  of  the  body  below  A  BCD,  or  the  volume 
immersed  in  a  state  of  rest,  and  the  other    that     comprised   between 


MECHANICS    OF    FLUIDS.  301 

ABCD  and  tlic  plane  uf  floatation  A' L'  C  D\  when  the  body  is  in 
motion.  Denote  by  rj  D  V  z\  the  value  of  the  first,  in  -which  z' 
denotes  the  variable  distance  H F^  of  the  centre  of  gravity  7/,  of 
the  volume  F,  from  the  plane  of  floatation  A'  B'  C  D'.  And  repre- 
senting for  the  moment  by  h  the  value  of  the  integral  I  zdV^  com- 
prehonded  between  the  planes  .1 /?  C'Z>  and  A' D'  C  D',  g  D  h  \\\\\ 
be    the    second   part;  and  Equation  (447)  becomes 


/ 


«2  JJ/  =  2y.  J/2-,  -  2gDVz'  -2gDh  +  C.  ■  ■  (440) 

The  line  GIf,  being  perpendicular  to  the  [ilane  ABCD,  the  angle 
which  it  makes  with  the  line  G  B  Is  equal  to  6,  and  denoting  the  dis- 
tance  G  H  by  a,  we  have 

2"^  =  2'  ±  a  cos  ^  ; 

the  upper  sign  being  taken  when  the  point  G  is  below  the  point 
H,  and  the  lower  when  it  is  above.  This  value  reduces  Equation 
(449)    to 

Ju^dM  =  ±2r/  D  Vacos6  —  2r;  Bh  -{-  C.    •    •   ■  (450) 

Let  us  now  find  the  integral  /;.  For  this  purpose,  conceive  the 
area  ABCD  to  be  divided  into  indefinitely  small  elements  denoted 
by  d\  and  let  these  be  projected  upon  the  plane  of  floatation, 
A'B'C'D'.  The  projecting  surflices  will  divide  the  volume  com- 
pressed  between  these  two  sections  into  an  indefinite  number  of 
vertical  elementary  prisms,  and  these  being  cut  by  a  series  of  hori- 
zontal planes  indciinitely  near  each  other,  will  give  a  series  of  ele- 
mentary volumes,  each  of  Avhich  will  be  denoted  by  d  V,  and  we 
shall   have 

d  V  =z  dz  .  dX.  cos  d  ; 

whence,  for  a    single  elementary  vertical    prism, 

fzdV  -  J  zdz.dX.  cos  6  =  i  (2)2  .cos6  .dX; 
in  which  (;)  denotes  the  mean  altitude  of  the  prism,  and  consequently 

h  =  I  cos  t)  .  f{zy  .  d  X, 
which   must   be    extended    to    ejnbrace    the    entire    surface  ABCD. 


302  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

The  value  of  (z)  is  composed  of  two  parts,  viz. :  one  comprised 
between  the  parallel  sections  A' B' C  D'  and  AB"CD",  and  which 
has  been  denoted  by  ^ ;  the  other  comprised  between  the  base  d  X 
and  the  second  of  these  planes,  and  which  is  equal  to  I .  sin  &,  de- 
noting by  /  the    distance  of  dX  from    the  intersection  A  C ;    whence, 

(.)  =^  +  ^sin^, 

in  which  I  will  be  positive  or  negative  according  as  dX  happens  to 
be  below  or  above  the  plane  AB"  C D".  Substituting  this  in  the 
value  of  A,  and  recollecting  that  ^  and  &  are  constant  in  the  inte- 
gration, we   find 

/t  =  i  2;2  _  cos  (5 .  fd  X  +  sin  &  cos&  J  IdX -\-  ^  sin2  ^  .  cos  ^  JP  d  X. 

Denote   by    b    the   area  of  A B  C D,  or   the  value   of  J  dX.     The 
line  A  C  passing  through  the  centre  of  gravity  of  A  B  C D,   we  have 
fldX  =  0.      And    denoting    by    k\  the    principal   radius  of  gyration 
of  the    surface  h,  in   reference    to    the   axis  A  C, 


fpdx  =  bk^^, 


in    which  the   value  of  l\    is    dependent   upon    the  figure    and    extent 

of  the  surface  A  B  C  D,  and  upon  the  position  of  the  line  AC. 
Whence, 

h  =  lb.  cos  6  (^2  +  A-,2  sin2  6).  ....     (451) 

Taking 

sin  6  =:  6  —  -— r.  +  «Scc ;     cos  ()  =  1  —  -— -  +  &c. 

Neglecting  all  the  terms  of  the  third  and  higher  orders,  substitut- 
ino-  the  value  of  h,  and  then  in  Equation  (450)  we  find,  after  trans- 
posing and    includuig    the    term    ±2(/D  Va,  in  the  constant   C, 

ftt\dM+  gB^b^^  +  (5  V  ±  Va)  02]=  C.  .  •  .(452) 

Now  the  value  of  the  constant  C  depends  upon  the  initial  values 
of  u,  d,  and  ^ ;  but  these  by  hypothesis  are  very  small ;  hence  (7, 
must   also   be  very    small.     As  long  as  the  second  term  of  the  first 


MECHAXICS     OF     FLUIDS.  303 

member  is  positive,  /  u-  d  M  must  remain  very  small,  since  it  is  essen- 
tially positive  itself,  and  being  increased  by  a  positive  quantity, 
the  sum  is  very  small.  Hence  ^  and  t'  must  remain  very  small. 
But  when  the  second  term  is  negative,  which  can  only  be  when 
bk^^  ±  Va,  is  negative  and  greater  than  b^'^,  the  value  ofJu^d3f 
may  increase  indefinitely ;  for,  being  diminished  by  a  quantity  that 
increases  as  fast  as  itself,  the  difference  may  be  constant  and  very 
small.  Hence,  ^  and  6  may  increase  more  and  more  afler  the 
body    is    abandoned  to  itself,    and   fmally   it   may  overturn. 

The  stability  of  the  equilibrium  depends,  therefore,  upon  the  sign 
(»f  i/i-^2  +  Yfi-^  ([iQ.  equilibrium  is  always  stable  when  this  quantity  is 
positive;  it  is  unstable  when  it  is  negative  and  greater  than  b  ^^. 
The  value  of  b  k/^  =  I  PdX,  must  always  be  positive,  since  all  its 
elements  are  positive ;  the  value  of  ±  Va  becomes  negative  when 
the  centre  of  gravity  of  the  body  is  above  that  of  the  .displaced 
fluid,  in    which    case    the  stability    requires   that 

iZ-,2>  Va,     or,  V"  >-f' 

When  the  centre  of  gravity  of  the  body  is  below  that  of  the  dis- 
placed  fluid,  the   sign    of    Va   is   positive. 

Whence  we  conclude  that  the  equilibrium  of  a  body  floating  at 
the  surface  of  a  heavy  fluid,  will  be  stable  as  long  as  the  centre 
of  gravity  of  the  body  is  below  that  of  the  displaced  fluid;  that 
it  will  also  be  stable  about  all  lines  A  C,  with  reference  to  which 
the  principal  radius  of  gyration  of  the  section  of  the  body  by  the 
plane  of  floatation  squared,  is  greater  than  the  volume  of  the  dis- 
placed fluid  multiplied  by  the  distance  between  the  centres  of 
gravity  of  the  displaced  fluid  and  that  of  the  body,  when  the  latter 
is  in  equilibrio,  divided  by  the  area  of  the  section  of  the  body 
by  the  plane  of  floatation.  When  this  condition  is  not  fulfilled,  the 
equilibrium  will  be  unstable.  A  ship  whose  centre  of  gravity  is 
above  that  of  the  water  she  displaces,  may  overturn  about  her  longer, 
but  not  about  her  shorter  axis, 

5^272. A  line  B K  tlirough  the  centre  of  gravity   G  of  the  body, 


304: 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


and  which  is  vertical  when  the  body  is  in  eqiiilibi'io,  is  called  a  line 
of  rest.  A  vertical  line  H'  M 
through  the  centre  of  gravity 
//'  of  the  displaced  fluid,  is 
called  a  line  of  sv2rport.  The 
point  il/,  in  which  the  line  of 
support  cuts  the  line  of  rest, 
is  called  the  metacentre.  The 
body  will  be  in  equilibrio 
when  the  line  of  rest  and  of 
support  coincide.  Tlie  equi- 
librium will  be  stable  if  the  metacentre  fall  above  the  centre  of 
gravity  ;    unstable   if  below. 

g273. — When  the  equilibrium  is  stable,  and  the  body  is  disturbed 
and  then  abandoned  to  the  action  of  its  own  weight  and  that  of 
the  fluid  pressure,  it  will,  in  its  efforts  to  regain  its  place  of  rest, 
oscillate   about   this  position,  and  finally  come   to    rest. 

The    circumstances  of  those    oscillations  about   the  centre  of  gravity 
of  the   body  will   readily  result  from  Equations  (445). 

SPECIFIC     GKAYITT. 


§274. — The  specific  gravittj  of  a  body,  is  the  Aveight  of  so  much 
of  the   body,  as  would    be    contained  under   a    unit    of   volume. 

It  is  measured  by  the  quotient  arising  from  dividing  the  weight 
of  the  body  by  the  weight  of  an  equal  volume  of  some  other  sub- 
stance, assumed  as  a  standard  ;  for  the  ratio  of  the  weights  of  equal 
volumes  of  two  bodies  being  always  the  same,  if  the  unit  of  volume 
of  each  be  taken,  and  one  of  the  bodies  become  the  standard,  its 
weight   will    become    the   unit  of  weight. 

The  term  density  denotes  the  degree  of  proximity  among  the 
particles  of  a  body.  Thus,  of  two  bodies,  that  will  have  the  greater 
density  which  contains,  under  an  equal  volume,  the  greater  number 
of  particles.  The  force  of  gravity  acts,  within  moderate  limits, 
equally    upon    all    elements    of    matter.      The    weight   of  a    substance 


MECIIAMCS     OF     FLUIDS.  305 

is,  therefore,  directly  proportional  to  its  density,  and  the  ratio  of 
the  weights  of  equal  volumes  of  two  bodies  is  equal  to  the  ratio 
of  their  densities.  Denote  the  weight  of  the  first  by  W,  its  density 
by  Z>,  its  volume  by    V,  and   the   force    of  gravity    by  y,  then  will 

W^ff.D.V; 

and  denoting  the  like  elements  of  the  other  body  by  W^ ,  D,  and 
V , ,  we   have 

Dividing    the   first   by  the  second. 


w 

ffDV          DV 

^K 

-  gD,V,  -   D,V, 

and   making  the  volumes 

equal, 

W        D 

w,  ~  d'  ' 

(453) 


Now  suppose  the  body  whose  weight  is  W^  to  be  assumed  as  the 
standard  botli  for  specific  gravity  and  density,  then  will  D^  be  unity, 
and 

W 

^=w:-^ (-^^^^ 

in  which  S  denotes  the  specific  gravity  of  the  body  whose  density 
is  D ;  and  from  which  we  see,  that  when  specific  gravities  and 
densities  are  referred  to  the  same  substance  as  a  standard,  the 
numbers  wliieh   express    the    one  will    also   express    the    other. 

§275. — Bodies  present  themselves  under  every  variety  of  condi- 
tion—gaseous, liquid,  and  solid ;  and  in  every  kind  of  shape  and  of 
all  sizes.  The  determination  of  their  .specific  gravity,  in  every  in- 
stance, depends  upon  our  ability  to  find  the  weight  of  an  equal 
volume  of  the  standard.  When  a  solid  is  immersed  in  a  fiuid,  it 
loses  a  portion  of  its  weight  equal  to  that  of  the  displaced  fluid. 
The  volume  of  the  body  and  that  of  the  displaced  fluid  are  equal. 
Hence  the  weight  of  the  body  in  vacuo,  divided  by  its  loss  of 
weight  when  immersed,  will  give  the  ratio  of  the  weights  of  equal 
volumes  of  the   body  and  fluid ;   and    if  the    latter   be    taken    as    the 

20 


306  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

standard,  and  the  loss  of  weight  he  made  to  occupy  the  denomi- 
nator, this  ratio  becomes  the  measure  of  the  specific  gravity  of  the 
body  immersed.  For  this  reason,  and  in  view  of  the  consideration 
that  it  may  be  obtained  pure  at  all  times  and  places,  luater  is 
assumed  as  the  general  standard  of  specific  gravities  and  densities 
for  all  bodies.  Sometimes  the  gases  and  vapors  are  referred  to 
atmospheric  air,  but  the  specific  gravity  of  the  latter  being  known 
as  referred  to  water,  it  is  very  easy,  as  we  shall  presently  see,  to 
pass  from  the  numbers  which  relate  to  one  standard  to  those  that 
refer    to    the    other, 

g  276. — But  water,  like  all  other  substances,  changes  its  density  with 
its  temperature,  and,  in  consequence,  is  not  an  invariable  standard. 
It  is  hence  necessary  either  to  employ  it  at  a  constant  temperature, 
or  to  have  the  means  of  reducing  the  apparent  specific  gravities,  as 
determined  by  means  of  it  at  diflerent  temperatures,  to  what  they 
would  have  been  if  the  water  had  been  at  the  standard  temperature. 
The  former  is  generally  impracticable;    the  latter  is  easy. 

Let  D  denote  the  density  of  any  solid,  and  S  its  specific  gravity, 
as  determined  at  a  standard  temperature  corresponding  to  which  the 
density  of  the  water  is  D,.     Then,  Equation  (453), 

Again,  if  S'  denote  the  specific  gravity  of  the  same  body,  as  indi- 
cated by  the  water  when  at  a  temperature  different  from  the  stan- 
dard,  and  corresponding  to  which  it  has  a  density  Z),^,  then  will 

Dividing  the  first  of  these  equations  by  the  second,  we  have 

S'         D/ 
whence, 

S=  S'--^; (455) 

and   if  the   density  J)^  ,  be  taken  as  unity, 

S=S'-D,,. (456) 


MECHANICS     OF    FLUIDS.  ^307 

That  is  to  say,  the  specific  gravitij  of  a  hodij  as  determined  at  the 
standard  temperature  of  the  water,  is  cqiial  to  its  specific  gravitij  deter- 
mined at  any  other  temperatiire,  multiplied  hy  the  density  of  the 
water  corresp)onding  to  this  temperature,  the  density  at  the  standard 
temperature   being  regarded  as  unity. 

To  make  this  rule  practicable,  it  becomes  necessary  to  find  the 
relative  densities  of  water  at  different  temperatures.  For  this  pur- 
pose, take  any  metal,  say  silver,  that  easily  resists  the  chemical 
action  of  water,  and  Avhose  rate  of  expansion  for  each  deforce  of 
Fahr.  thermometer  is  accurately  known  from  experiment;  give  it 
the  form  of  a  slender  cylinder,  that  it  may  readily  conform  to  the 
temperature  of  the  Mater  when  immersed.  Let  the  length  of  the 
cylinder  at  the  temperature  of  33°  Fahr.  be  denoted  by  I,  and  the 
radius   of  its   base   by   ml;    its  volume   at   this  temperature  will  be 

•Tf  m'^P  X  I  =:  -rf  m-  P- 

Let  n  I  be  the  amount  of  expansion  in  length  for  each  decree  of 
the  thermometer  above  32°.  Then,  fur  a  temperature  denoted  by 
t,  will    the   whole    expansion   in  length  be 

nl  X  {t  —  32°), 

and  the  entire  length  of  the  cylin- 
der  will   become 

/-f/i/(;-32°)  =  /[l  +  7i  (^-32°)]; 

which,  substituted  for  I  in  the  first 
expression,  will  give  the  volume 
fi)r  the  temperature  /,  equal  to 

ifm^Pll  ^  n{t  —  32°)] \ 

The  cylinder  is  now  weighed  in 
vacuo  and  in  the  water,  at  differ- 
ent  temperatures,  varying  from  32° 
upward,  through  any  desirablp  range, 
say  to  one  hundred  degrees.  The 
temperature  at  each  process  being 
substituted  above,  gives  the  volume 


1 i 

of  the   displaced   fluid ;    the   weight   of    the   displaced  fluid   is 


knc 


308 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


from  the  loss  of  weight  of  the  cylinder.  Dividing  this  weight  by 
the  volume,  gives  the  weight  of  the  unit  of  volume  of  the  water  at 
the  temperature  t.  It  was  found  by  Starnpfer,  that  the  weight  of 
the  unit  of  volume  is  greatest  when  the  temperature  is  38°.75  Fah- 
renheit's scale.  Taking  the  density  of  the  warer  at  this  temperature 
as  unity,  and  dividing  the  weight  of  the  unit  of  volume  at  each  of 
the  other  temperatures  by  the  weight  of  the  unit  of  volume  at  this, 
38°.75,  Table  II  will  result. 

The  column  under  the  head  P",  will  enable  us  to  determine  how 
much  the  volume  of  any  mass  of  water,  at  a  temperature  t,  exceeds 
that  of  the  same  mass  at  its  maximum  density.  For  this  purpose, 
we  have  but  to  multiply  the  volume  at  the  maximum  density  by 
the   tabular   number   corresponding  to  the  given  temperature. 

g  277. — Before  proceeding  to  the  practical  methods  of  finding  the 
specific  gravity  of  bodies,  and  to  the  variations  in  the  processes 
rendered  necessary  by  the  peculiarities  of  the  different  substances, 
it  will  be  necessary  to  give  some  idea  of  the  best  instruments  em- 
ployed for  this  purpose.  These  are  the  Hydrostatic  Balance  and 
Nicholsoii's  Hydrometer. 

The  first  is  similar  in  principle  and  form  to  the  common  balance. 
It  is  provided  with  numerous 
weights,  extending  through  a 
wide  range,  from  a  small 
fraction  of  a  grain  to  several 
ounces.  Attached  to  the  un- 
der surface  of  one  of  the 
basins  is  a  small  hook,  from 
which  may  be  suspended 
any  body  by  means  of  a 
thin  platinum  wire,  horse- 
hair, or  any  other  delicate 
thread  that  will  neither  absorb 
nor  yield  to  the  chemical  ac- 
tion of  the  fluid    in    which    it   may   be   desirable  to  immerse  it. 

Nicholson's  Hydrometer  consists  of  a  hollow  metalic  ball  A,  through 


MECHANICS     OF    FLUIDS. 


309 


the  centre  of  which  passes,  a  metallic  wire,  prolonged  in  both  di- 
rections beyond  the  surface,  and  supporting 
at  either  end  a  basin  B  and  B'.  The 
concavities  of  these  basins  are  turned  in 
the  same  direction,  and  the  basin  B'  is 
made  so  heavy  that  ^vhen  the  instrument 
is  placed  in  water  the  stem  C  C  shall  be 
vertical,  and  a  weight  of  500  grains  being 
placed  in  the  basin  B^  the  whole  instrument 
will  sink  till  the  upper  surface  of  distilled 
water,  at  the  standard  temperature,  comes  to 
a   point    C  marked   on   the  upper  stem  near 

its   middle.     This   instrument   is    provided    with    weights    similar    to 
those   of  the   Hydrostatic   Balance. 

g278. — (1).  If  the  body  he  solid,  insoluble  in  water,  and  will  sink 
in  that  fluid,  attach  it,  by  means  of  a  hair,  to  the  hook  of  the 
basin  of  the  hydrostatic  balance  ;  counterpoise  it  by  placing  weights 
in  the  opposite  scale  ;  now  immerse  the  body  in  water,  and  restore 
the  equilibrium  by  placing  weights  in  the  basin  above  the  body, 
and  note  the  temperature  of  the  water.  Divide  the  w^eights  in  the 
basin  to  which  the  body  is  not  attached  by  those  in  the  basin  to 
which  it  is,  and  multiply  the  quotient  by  the  density  corresponding 
to  the  temperature  of  the  water,  as  given  by  the  table ;  the  result 
will    be   the   specific  gravity. 

Thus  denote  the  specific  gravity  by  S,  the  density  of  the  water 
by  Z>,, ,  the  weight  in  the  first  case  by  W,  and  that  in  the  scale 
above   the  solid  by  w,  then  will 

W 


(2).  If  the  bodij  be  insoluble,  but  will  not  sink  in  ivater,  as  would 
be  the  case  with  most  varieties  of  wood,  wax,  and  the  like,  attach 
to  it  some  body,  as  a  metal,  whose  weight  in  the  air  and  loss  of 
weight  in  the  water  are  previously  found.  Then  proceed,  as  in  the 
case  before,  to  find  the  weights  which  will  counterpoise  the  com- 
poimd    in    air  and   restore   the    equilibrium  of  the  balance  when  it  is 


310  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

immersed  in  the  water.  From  the  weight  of  the  compound  in  air, 
subtract  that  of  the  denser  body  in  air ;  from  the  loss  of  weight 
\of  the  compound  in  water,  subtract  that  of  the  denser  body ; 
divide  the  first  difference  by  the  second,  and  multiply  by  the  density 
of  the  water  answering  to  its  temperature,  and  the  result  will  be 
the   specific   gravity  sought. 

Example. 

grs. 

A  piece  of  wax  and  copper  in  air  =  438      =z  W  -{-  W\ 
Lost   on   immersion  in  water  -     -     =    95,8  =  lo  -{-   w', 

Copper  in  air =  388      =  W, 

Loss  of  copper  in  water      -     •     -     =    44,2  =  w'. 

Then 

W+  W  -  W  =  438  -  388   =  50,     =  TF, 
w  +  w'  —  w'   =  95,8  —  44,2  =  51,6  =  w. 

Tempei'ature  of  water  43°,25, 

D,,  -  0,999952, 

W  50 

S  =  D„x  —   =  0,999952  x    ^^  =  0,968. 
w  51, o 

(3).  If  the  lody  readily  dissolve  in  water,  as  many  of  the  salts, 
sugar,  &c.,  find  its  apparent  specific  gravity  in  some  liquid  in  which 
it  is  insoljible,  and  multiply  this  apparent  specific  gravity  by  the 
density  or  specific  gravity  of  the  liquid  referred  to  water  as  its 
maximum  density  as  a  standard ;  the  j)roduct  will  be  the  true  specific 
gravity. 

If  it  be  inconvenient  to  provide  a  liquid  in  which  the  solid  is 
insoluble,  saturate  the  water  with  the  substance,  and  find  the  appa- 
rent specific  gravity  with  the  water  thus  saturated.  Multiply  this 
apparent  specific  gravity  by  the  density  of  the  saturated  fluid,  and 
the  product  will  be  the  specific  gravity  referred  to  the  standard. 
This  is  a  common  method  of  finding  the  specific  gravity  of  gunpow- 
der, the  water   being    saturated  with  nitre. 

(4).  If  the  body  he  a  liquid,  select  some  solid  that  will  resist  its 
chemical   action,    as   a   massive    piece    of   glass    suspended   from   fine 


MECHAXICS     OF     FLUIDS.  311 

platinum  wire ;  \Yeigh  it  in  air,  then  iu  water,  and  finally  in  the 
liquid ;  the  differences  between  the  first  weight  and  each  of  the 
latter,  will  give  the  weights  of  equal  volumes  of  water  and  the 
liquid.  Divide  the  weight  of  the  liquid  by  that  of  the  water,  and 
the  quotient  will  be  the  specific  gravity  of  the  liquid,  provided  the 
temperature  of  water  be  at  the  standard.  If  the  water  have  not 
the  standard  temperature,  multiply  this  apparent  specific  gravity  by 
the  tabular  density  of  the  water  corresponding  to  the  actual  tem- 
perature. 

Example. 

grs. 

Loss  of  glass  in  water  at  41°,  150      =  w\ 
"  "         sulphuric  acid,  277,5  =  w, 

277  5 
S  =  -— ^  X  0,999988  =  1,85. 

(5).  If  the  body  be  a  gas  or  vapor,  provide  a  large  glass  flask- 
shaped  vessel,  weigh  it  when  filled  with  the  gas  ;  withdraw  the  gas, 
which  may  be  done  by  means  to  be  explained  presently,  fill  Avith 
water,  and  weigh  again  ;  finally,  withdraw  the  water  and  exclude  the 
air,  and  weigh  again.  This  last  weight  subtracted  from  the  first, 
will  give  the  weight  of  the  gas  that  filled  the  vessel,  and  subtracted 
from  the  second  will  give  the  weight  of  an  equal  volume  of  water; 
divide  the  weight  of  the  gas  by  that  of  the  water,  and  multiply 
by  the  tabular  density  of  the  water  answering  to  the  actual  tem- 
perature of  the  latter ;  the  result  will  be  the  specific  gravity  of 
the   gas. 

The  atmosphere  in  which  all  these  operations  must  be  performed, 
varies  at  different  times,  even  during  the  same  day,  in  respect  to 
temperature,  the  weight  of  its  column  which  presses  upon  the  earth, 
and  the  quantity  of  moisture  or  aqueous  vapor  it  contains.  Tliat  is 
to  say,  its  density  depends  upon  the  state  of  the  thermometer,  barom- 
eter, and  hygrometer.  On  all  these  accounts  corrections  must  be 
made,  before  the  specific  gravity  of  atmospheric  air,  or  that  of  any 
gas  exposed  to  its  pressure,  can  be  accurately  determined.  The  prin- 
ciples according  to  which  these  corrections  are  made,  will  be  discussed 
when  we  come  to  treat  of  the  properties  of  elastic   fluids. 


n2 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


To  find  the  specific  gravity  of  a  solid  by  means  of  Nicholson's 
Hydometer,  place  the  instrument  in  water,  and  add  weights  to  the 
upper  basin  until  it  sinks  to  the  mark  on  the  upper  stem;  remove 
the  weights  and  place  the  solid  in  the  upper  basin,  and  add  weights 
till  the  hydrometer  sinks  to  the  same  point;  the  difference  between 
the  first  weights  and  those  added  with  the  body,  will  give  the 
weight  of  the  latter  in  air.  Take  the  body  from  the  upper  basin, 
leaving  the  weights  behind,  and  place  it  in  the  lower  basin ;  add 
weights  to  the  upper  basin  till  the  instrument  sinks  to  the  same  point 
as  before,  the  last  added  weights  will  be  the  weight  of  the  water 
displaced  by  the  body  ;  divide  the  weight  in  air  by  the  weight  of 
the  displaced  water,  and  multiply  the  quotient  by  the  tabular  density 
of  the  water  answering  to  its  actual  temperature  ;  the  result  will  be 
the  specific  gravity  of  the  solid. 

To  find  the  specific  gravity  of  a  fluid  by  this  instrument,  immerse 
it  in  water  as  before,  and  by  weights  in  the  upper  basin  sink  it  to 
the  mark  on  the  upper  stem  ;  add  the  weights  in  the  basin  to  the 
weight  of  the  instrument,  the  sum  will  be  the  weight  of  the  dis- 
placed water.  Place  the  instrument  in  the  fluid  whose  specific  gravity 
is  to  be  found,  and  add  weights  in  the  upper  basin  till  it  sinks  to 
the  mark  as  before ;  add  these  weights  to  the  weight  of  the  instru- 
ment, the  sum  will  be  the  weight  of  an  equal  volume  of  the  fluid ; 
divide  this  weight  by  the  weight  of  the 
water,  and  multij^ly  by  the  tabular  density 
corresponding  to  the  temperature  of  the 
water,  the  result  will  be  the  specific  gravity. 

§  279. — Besides  the  hydrometer  of  Nichol- 
son, which  requires  the  use  of  weights,  there 
is  another  form  of  this  instrument  which  is 
employed  solely  in  the  determination  of  the 
specific  gravities  of  liquids,  and  its  indications 
are  given  by  means  of  a  scale  of  equal  parts. 
It  is  called  the  Scale- Areometer.  It  consists, 
generally,  of  a  glass  vial-shaped  vessel  A,  ter- 
minating at  one  end  in  a  long  slender  neck  C, 
to  receive  the  scale,  and   at   the    other   in   a 


MECHANICS     OF     FLUIDS.  313 

small  globe  i?,  filled  with  some  heavy  substance,  as  lead  or  mercury 
to  keep  it  upright  when  immersed  in  a  fluid.  The  application  and 
use  of  the  scale  depend  upon  this,  that  a  body  floating  on  the  surface 
of  diflerent  liquids,  will  sink  deeper  and  deeper,  in  proportion  as  the 
density  of  the  fluid  approaches  that  of  the  body  ;  for  when  the  body 
is  at  rest  its  weight  and  that  of  the  displaced  fluid  must  be  equal. 
Denoting  the  volume  of  the  instrument  by  V,  that  of  the  dis- 
placed fluid  by  F',  the  density  of  the  instrument  by  D,  and  that 
of  the   fluid   by   £>',  we   must   always   have 

in  which  ff  denotes  the  force  of  gravity,  the  first  member  the  weight 
of  the  instrument,  and  the  second  that  of  the  displaced  fluid.  Divi- 
ding both  members  by  J)'  V,  and  omitting  the  common  factor  g 
we   have 

I)    _   V 

1)'  ~  y 

In  which,  if  the  densities  be  equal,  the  volumes  must  be  equal ; 
if  the  density  B'  of  the  fluid  be  greater  than  Z>,  or  that  of  the 
solid,  the  volume  V  of  the  solid  must  be  greater  than  V,  or  that 
of  the  displaced  fluid ;  and  in  proj)ortion  as  D'  increases  in  i-cspect 
to  I),  will  V  diminish  in  respect  to  V ;  that  is,  the  solid  will 
rise  higher  and  higher  out  of  the  fluid  in  proportion  as  the  den- 
sity of  the  latter  is  increased,  and  the  reverse.  The  neck  C  of 
the  vessel  should  be  of  the  same  diameter  throughout.  To  estab- 
lish the  scale,  the  instrument  is  j)laeed  in  distilled  water  at  the 
standard  temperature,  and  when  at  rest  the  place  of  the  surflice 
of  the  water  on  the  neck  is  marked  and  numbered  1  ;  the  instru- 
ment is  then  placed  in  some  heavy  solution  of  salt,  whose  specific 
gravity  is  accurately  known  by  means  of  the  Hydrostatic  Balance, 
and  when  at  rest  the  place  on  the  neck  of  the  fluid  surface  is  attain 
marked  and  characterized  by  its  appropriate  number.  The  same  pro- 
cess being  repeated  fur  rectified  alcoliDl,  will  give  another  point 
towards  the  opposite  extreme  of  the  scale,  which  may  be  completed 
by  graduation.  „ 


314         ELEMENTS     OF     ANALYTICAL    MECHANICS. 

To  use  this  instrument,  it  will  be  sufficient  to  immerse  it  in  a 
fluid  and  take  the  number  on  the  scale  which  coincides  with  the 
surface. 

To  ascertain  the  circumstances  which  determine  the  sensibility 
both  of  the  Scale- Areometer  and  Nicholson's  Hydrometer,  let  s  de- 
note the  specific  gravity  of  the  fluid,  c  the  volume  of  the  vial,  I  the 
length  of  the  immersed  portion  of  the  narrow  neck,  r  its  semi-diame- 
ter, and  10  the  total  weight  of  the  instrument.  Then  will  *  ?-2,  denote 
the  area  of  a  section  of  the  neck,  and  *  r"^  I,  the  volume  of  fluid  dis- 
placed by  the  immersed  part  of  the  neck.  The  weight,  therefore,  of 
the  whole  fluid  displaced  by  the  vial  and  neck  will  be    • 

s  c  +  sit  r-l ; 

but  this  must  be  equal  to  the  weight  of  the  instrument,  whence, 

w  ^=  s{c  -\-  itr"^  I), 
from  which  we  deduce, 


c  -\-  'Tfr'^V 

-            10    —    SC  ,,^rr\ 

1  =  5— •i'i^^) 

If  r^s 

Now,  immersing  the  instrument  in  a  second  fluid  whose  specific  gravi- 
ty is  s',  the  neck  will  sink  through  a  distance  l',  and  from  the  last 
equation  we  have 

70    —    s' C 


I' 


,.2  „'      ' 


subtracting  this  equation  from  that  above  and  reducing,  we  And 

The  difference  I  —  I'  is  the  distance  between  two  points  on  the  scale 
which  indicates  the  difference  s'  —  s  of  specific  gravities,  and  this 
we  see  becomes  longer,  and  the  instrument  more  sensible,  therefore, 
in  proportion  as  lo  is  made  greater  and  r  less.  Whence  Ave  con- 
clude that  the  Areometer  is  the  more  valuable  in  proportion  as  the 
vial  portion  is  made  larger  and  the  neck  smaller. 


MECHANICS     OF    FLUIDS.  ^  316 

If  the  specific  gravity  of  the  fluid  remain  the  same,  wliich  is  the 
case  with  Nicholson's  Hydrometer,  and  it  becomes  a  question  to 
know  the  eflect  of  a  small  weight  added  to  the  instrument,  denote 
this    weight  by  w\  then  will  E(|uation  (457)  become 

w  -{■  to'  —  s  c 
ir  r^  s         ' 

subtracting  from  this  Ecj[uati(ni   (I'jT),  wc  find 

From  which  "we  see  that  the  narrower  the  upper  stem  of  Nicholson's 
instrument,  the  greater  its  sensibility. 

The  knowledge  of  the  specific  gravities  or  densities  of  difierent 
substances,  Table  III,  is  of  great  importance,  not  only  for  scientific 
purposes,  but  also  for  its  application  to  many  of  the  useful  arts. 
This  knowledge  enables  us  to  solve  such  problems  as  the  follow- 
ing, viz.  : — 

1st.  The  weight  of  any  substance  may  be  calculated,  if  its  volume 
and    specific   gravity  be    known. 

2d.  The  volume  of  any  body  may  be  deduced  from  its  specific 
gravity  and  weight.     Thus  we   have   always 

W  =  ffBV; 

in  which  ff  is  the  force  of  gravity,  D  the  densit)'',  V  the  volume, 
and  W  the  weight,  of  wliich  the  unit  of  measure  is  the  weight  of 
a   unit    of  volume    of  water  at   its   maximum    density. 

flaking  D  and   V  equal   to   unity,  this  equation  becomes 

but  if  the  density  be  one,  the  substance  must  be  water  at  3S°,75 
Fahr.  The  weight  of  a  cubic  foot  of  water  at  G0°  is  G2,5  lbs.,  and, 
therefore,  at  3S°,75,  it  is 

lbs:. 

^^'^      =  62,550  ; 


0,99914 

whence,  if  the   volume   be   expressed    in    cubit   feet, 


W  =  G2,55G  X  DV (458) 


316 


ELR-MENTS     OF    A^'ALYTICAL    MECHANICS. 


in  which    W  is  expressed  in    pounds  ;    and  if  the  unit    of  volume  be 
a   cubic   inch, 

C2,550 


W 


Also, 


1728 
V  = 


DV  =  0,036201  Z>  V, 

w: 


V.  = 


lbs. 

62,556  .  J) 
W. 

lbs. 

0,036201  .  D 


(459) 
(460) 

(461) 


Example  1.— Required  the  weight  of  a  block  of  dry  fir,  containing 
50  cubic  inches.  The  specific  gravity  or  density  of  dry  fir  is  0,555, 
and    F  =  50  ;    substituting  these   values  in  Equation  (459), 

lbs. 

W  —  0,030201   X  0,555  X  50  =  1,00457. 
ExanvpJe  2.— How    manj    cubic   inches    are    there    in   a     12-pound 
cannon-ball  %     Here    W  is    12   pounds,  the   mean   specific   gravity    of 
cast   iron  is  7,251,  which,  in  Equation  (461),  give 

12 


V.  = 


0,036201   X  7,251 


=  45,6. 


ATMOSPHEEIC   PKESSUEE. 

8  280.— The  atmosphere  encases,  as  it  were,  the  whole  earth.  It 
has  weight,  else  the  repulsive  action  among  its  own  pai-ticles  would 
cause  it  to  expand  and  extend  itself  through  space.  The  weight  of 
the  upper  stratum  of  the  atmosphere  is  in  equilibrio  with  the  re- 
pulsive action  of  the  strata  below  it,  and  this  condition  determines 
the    exterior  limit. 

Since  the  atmosphere  has  weight,  it  must 
exert  a  pressure  upon  all  bodies  within  it. 
To  illustrate,  fill  with  mercury  a  glass  tube, 
about  32  or  33  inches  long,  and  closed  at 
one  end  by  an  iron  stop-cock.  Close  the 
open  end  by  pressing  the  finger  against  it, 
and  invert  the  tube  in  a  basin  of  mercury; 
remove  the  finger,  the  mercury  will  not 
esca^^e,  but  remain    apparently   suspended,    at 


MECHANICS    OF    FLUIDS.  317 

the   level   of  the  ocean,  nearly   30   inches   above   the    surface   of  the 
mercury  in  the   basin. 

The  atmospheric  air  presses  on  the  mercury  with  a  force  sufficient 
to  maintain  the  quicksilver  in  the  tube  at  a  height  of  nearly  30 
inches  ;  Avhence,  the  intensity  of  its  2>ressure  must  be  equal  to  the  weight 
of  a  column  of  mercury  ivhose  base  is  equal  to  that  of  the  surface 
pressed  and  whose  altitude  is  about  30  inches.  The  force  thus  exerted, 
is  called  the  atmos2:>heric  pressure. 

The  absolute  amount  of  atmospheric  pressure  was  first  discovered 
by  Torricelli,  and  the  tubes  employed  in  such  experiments  are  called, 
on  tills  acc(5unt,  Torricellian  tubes,  and  the  vacant  space  above  the 
mercury  in   the  tube,  is  called   the  Torricellian  vacuum. 

The  pressure  of  the  atmosphere  at  the  level  of  the  sea,  support- 
ino-  as  it  does  a  column  of  mercury  30  inches  high,  if  we  suppose 
the  bore  of  the  tube  to  have  a  cross-section  of  one  square  inch, 
the  atmospheric  pressure  up  the  tube  will  be  exerted  upon  this 
extent  of  surflice,  and  will  support  30  cubic  inches  of  mercury. 
Each  cubic  inch  of  mercury  weighs  0,49  of  a  pound — say  half  a 
pound — from  which  it  is  apparent  that  the  surfaces  of  all  bodies,  at 
the  level  of  the  sea,  are  subjected  to  an  atmospheric  pressure  of  fifteen 
pounds   to  each  square  inch. 

BAROilETER. 

§281. — The  atmosphere  being  a  heavy  and  elastic  fluid,  is  com- 
pressed by  its  own  weight.  Its  density  cannot  be  the  same  through- 
out, but  diminishes  as  we  approach  its  upper  limit  where  it  is  least, 
being  greatest  at  the  surface  of  the  earth.  If  a  vessel  filled  with 
air  be  closed  at  the  base  of  a  high  mountain  and  afterwards  opened 
on  its  summit,  the  air  will  rush  out ;  and  the  vessel  being  closed 
again  on  the  summit  and  opened  at  the  base  of  the  mountain,  the 
air  will  rush  in. 

The  evaporation  which  takes  place  from  large  bodies  of  Avater, 
the  activity  of  vegetable  and  animal  life,  as  well  as  vegetable  decom- 
positions, throw  considerable  quantities  of  aqueous  vapor,  carbonic 
acid,  and    other    foreign    ingredients   temporarily    into    the    permanent 


318 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


portions  of  the  atmosphere.  These,  together  with  its  ever-varying 
temperature,  keep  the  density  and  elastic  force  of  the  air  in  a 
state  of  almost  incessant  change.  These  changes  are  indicated  by 
the  Barometer^  an  instrument  employed  to  measure  the  intensity  of 
atmospheric  pressure,  and  frequently  called  a  weather-glass^  because 
of  certain  agreements  found  to  exist  between  its  indications  and  the 
state    of  the    weather. 

The  barometer  consists  of  a  glass  tube  about  thirty-four  or  thirty- 
five  inches  long,  open  at  one  end,  partly  filled  with  distilled  mer- 
cury, and  inverted  in  a  small  cistern  also  containing  mercury.  A 
scale  of  equal  parts  is  cut  upon  a  slip  of  metal,  and  placed  against 
the  .tube  to  measure  the  height  of  the  mercurial  column,  the  zero 
being  on  a  level  with  the  surfiicc  of  the  mercury  in  the  cistern. 
The  elastic  force  of  the  air  acting  freely  upon  the  mercury  in  the 
cistern,  its  pressure  is  transmitted  to  the  interior  of  the  tube,  and 
sustains  a  column  of  mercury  whose  weight  it  is  just  sufficient  to 
counterbalance.  If  the  density  and  consequent  elastic 
force  of  the  air  be  increased,  the  column  of  mei'cury 
will  rise  till  it  attain  a  corresponding  increase  of 
weight;  if,  on  the  contrary,  the  density  of  the  air 
diminish,  the  column  will  foil  till  its  diminished 
weight  is   sufiicient    to   restore    the    equilibrium. 

In  the  Common  Barometer,  the  tube  and  its  cis- 
tern are  partly  inclosed  in  a  metallic  case,  upon 
which  the  scale  is  cut,  the  cistern,  in  this  case,  hav- 
ing a  flexible  bottom  of  leather,  against  Avhieh  a 
plate  a  at  the  end  of  a  screw  b  is  made  to  press, 
in  order  to  elevate  or  depress  the  mercury  in  the 
cistern  to   the  zero    of  the   scale. 

De  Luc's  Sij^hon  Barometer  consists  of  a  glass 
tube  bent  upward  so  as  to  form  two  unequal  par- 
allel legs :  the  longer  is  hermetically  sealed,  and 
constitutes  the  Torricellian  tube ;  the  shorter  is  open, 
and  on  the  surface  of  the  quicksilver  the  pressure 
of  the  atmosphere  is  exerted.  The  difference  be- 
tween   the   levels    in    the  longer   and   shorter   legs    is    the   baromfetric 


A 


MECHANICS    OF     FLUIDS. 


31& 


height.     The   most  convenient  and  practicable  way  of  measuring  this 
difference,    is    to    adjust    a    movable    scale    between 
the   two   legs,    so    that   its   zero    may   be    made    to 
coincide    with    the    level    of    the    mercury    in     the 
shorter    leg. 

Different  contrivances  have  been  adopted  to  ren- 
der the  minute  variations  in  the  atmospheric  pres- 
sure, and  consequently  in  the  height  of  the  barome- 
ter, more  readily  perceptible  by  enlarging  the  di- 
visions on  the  scale,  all  of  which  devices  tend  to 
hinder  the  exact  measurement  of  the  length  of  the 
column.  Of  these  we  may  name  Morland's  Diago- 
nal, and  Hook's  Wheel-Barometer,  but  especially 
-Huy gen's  Double-Barometer. 

The  essential  properties  of  a  good  barometer, 
are :  width  of  tube  ;  purity  of  the  mercury  ;  accu- 
rate graduation   of  the   scale;  and   a  good  vernier. 

§282. — The  barometer  may  be  used  not  only  to  measure  the 
pressure  of  the  external  air,  but  also  to  determine  the  density  and 
elasticity  of  pent-up  gases  and  vapors.  When  thus  employed,  it  is 
called  the  barometer- gauge.  In  every  case  it  will 
only  be  necessary  to  establish  a  free  connection 
between  the  cistern  of  the  barometer  and  the  vessel 
containing  the  fluid  whose  elasticity  is  to  be  indi- 
cated ;  the  height  of  the  mercury  in  the  tube, 
expressed  in  inches,  reduced  to  a  standard  tempera- 
ture, and  multiplied  by  the  known  weight  of  a 
cubic  inch  of  mercury  at  that  temperature,  will 
give  the  pressure  in  pounds  on  each  square  inch. 
In  the  case  of  the  steam  in  the  boiler. of  an  en- 
gine, the  upper  end  of  the  tube  is  sometimes  left 
open.  The  cistern  A  is  a  steam-tight  vessel,  partly 
filled  with  mercury,  a  is  a  tube  communicating 
with  the  boiler,  and  through  which  the  steam  flows 
and  presses  upon  the  mercury  ;  the  barometer  tube 
he,    opt'U    at    top,    reaehes    nearly    to    the    bottom    of    the    vessel    J, 


320 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


havinjr  attached  to  it  a  scale  -whose  zero  coincides  with  the  level 
of  the  quicksilver.  On  the  right  is  marlved  a  scale  of  inches,  and 
on  the  left   a   scale   of  atmospheres. 

If  a  very  high  pressure  were  exerted,  one  of  several  atmospheres 
for  example,  an  apparatus  thus  constructed  would 
require  a  tube  of  great  length,  in  which  case  Ma- 
riotle's  manometer  is  considered  preferable.  The  tube 
being  filled  with  air  and  the  upper  end  closed,  the 
surface  of  the  mercury  in  both  branches  will  stand 
at  the  same  level  as  long  as  no  steam  is  admitted. 
The  steam  being  admitted  through  d,  presses  on  the 
surface  of  the  mercury  a  and  forces  it  up  the  branch 
h  c,  and  the  scale  from  6  to  c  marks  the  force  of 
compression  in  atmospheres.  The  greater  width  of 
tube  is  given  at  a,  in  order  that  the  level  of  the 
mercury  at  this  point  may  not  be  materially  affected 
by  its  ascent  up  the  branch  be,  the  point  a  being  the  zero  of  the 
scale. 


g283. — Another  very  important  use  of  the  barometer,  is  to  find 
the  difference  of  level  between  two  places  on  the  earth's  surface,  as 
the  foot   and   top    of  a   hill    or    mountain. 

Since  the  altitude  of  the  barometer  depends  on  the  pressure  of 
the  atmosphere,  and  as  this  force  depends  upon  the  height  of  the 
pressing  column,  a  shorter  column  will  exert  a  less  pressure  than  a 
longer  one.  The  quicksilver  in  the  barometer  falls  w^hen  the  instru- 
ment is  carried  from  the  foot  to  the  top  of  a  mountain,  and  rises 
again  when  restored  to  its  first  position :  if  taken  down  the  shaft 
of  a  mine,  the  barometric  column  rises  to  a  still  greater  height.  At 
the  foot  of  the  mountain  the  whole  column  of  the  atmosphere,  from 
its  utmost  limits,  presses  with  its  entire  weight  on  the  mercury; 
at  the  top  of  the  mountain  this  weight  is  diminished  by  that  of 
the  intervening  stratum  between  the  two  stations,  and  a  shorter 
column  of  mercury   will    be    sustained   by    it. 

It  is  well  known  that  the  surface  of  the  earth  is  not  uniform, 
and  does  not,  in  consequence,  sustain   an  equal  atmospheric  j^ressure 


MECHANICS     OF    FLUIDS.  321 

at  its  different  points;  whence  the  mean  altitude  of  the  barometric 
column  will  vary  at  dilTorent  places.  This  furnishes  one  of  the 
best  and  most  expeditious  means  of  getting  a  profile  of  an  extended 
section  of  the  earth's  surface,  and  makes  the  barometer  an  instru- 
ment of  great  value  in  the  hands  of  the  traveller  in  search  of 
geographical   information. 

g  284. — To  fmd  the  relation  which  subsists  between  the  altitudes 
of  two  barometric  columns,  and  the  difference  of  level  of  the  points 
where  they  exist,  resume  Equation  (427).  The  only  extraneous  force 
acting  being  that  of  gravity,  we  have,  taking  the  axis  z  vertical, 
and  counting  z   positive   upwards, 

X  =  0 ;     r  =  0  ;    Z=-g. 
and  hence, 

^  =  Ce~  9 (462) 

Making  z  =  0,  and  denoting  the  corresponding  pressure  by  j9^,  we  find 

T.  =  0; 
and  dividing  the  last  Equation  by  this  one, 

P         -^± 

—  —e    p,  , 

whence,  denoting   the  reciprocal  of  the   common  modulus  by  J/, 

z  = log  — ^ (-loo) 

9  ""  P 

Denote  by  h^  and  A,  the  barometric  heights  at  the   lower   and  upper 

stations,  respectively,  then  will 

Pj_  _   h_. 

and  reducing  the  barometric  column  h  to  what  it  Avould  have  been 
had  the  temperature  of  the  mercury  at  the  upper  not  differed  from 
that  at   the  lower  station,  by  Equation  (394),  we  have 


P, 


h. 


■p    ~  h\\  ■\-{T-  T)  .0,0001001  ' 

in  which  T  denotes  the  temperature  of  the  mercury  at  the  lower  and 
T  that  at  the  upper  station. 


21 


322  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Moreovei',  Equation  (381), 

g  =  g'  {\-  0,002551  cos  2  4.)  ; 

in  which, 

/ 
g'  =  32,1808  =  force   of  gravity  at  the  latitude  of  45°. 

P 

Substituting   the  value  of  —  • ,    of  ^,  and  that  of  F,  as  given    by 

P 
Equation  (393),  in  Equation  (463),  we  find 

___MDJ,,   l  +  (^-32)0,00208  p  1  ] 

D,     '1-0,002551008  24.^  ^ La     i  +  (y'-r)o,oooiooi  J 

In  this  it  will  be  remembered  that  t  denotes  the  temperature  of 
the  air;  but  this  may  not  be,  indeed  scarcely  ever  is,  the  same  at 
both  stations,  and  thence  arises  a  difficulty  in  applying  the  formula. 
But  if  we  represent,  for  a  moment,  the  entire  factor  of  the  second 
member,  into  which  the  factor  involving  t  is  multiplied,  by  X,  then 
we   may  write 

2  =  [1  +  (/  _  32°)0,00208]  X. 

If  the  temperature  of  the  lower  station  be  denoted  by  i, ,  and  this 
temperature  be  the  same  throughout  to  the  upper  station,  then  will 

2^  =  [1  +  (/^  -  32°)  0,00208]  X. 

And  if  the  actual  temperature  of  the  iijrper  station  be  denoted  by  t\ 
and  this  be  supposed  to  extend  to  the  lower  station,  then  would 

s'  =  [1  +  {f  -  32°)  0,00208]  X. 

Now  if  t,  be  greater  than  t\  which  is  usually  the  case,  then  will  the 
barometric  column,  or  A,  at  the  upper  station,  be  greater  than  would 
result  from  the  temperature  t\  since  the  air  being  more  expanded, 
a  portion  which  is  actually  below  would  pass  above  the  upper 
station  and  press  upon  the  mercury  in  the  cistern  ;  and  because  h 
enters  the  denominator  of  the  value  -X",  z,  would  be  too  small. 
Again,  by  supposing  the  temperature  the  same  as  that  at  the  upper 
station  throughout,  then  would  the  air  be  more  condensed  at  the 
lower  station,  a  portion  of  the  air  would  sink  below  the  upper 
station  that  before  was  above  it,  and  would  cease  to  act  upon  the 
mercurial  column  h,  which  would,  in  consequence,  become  too  small ; 


MECHANICS    OF    FLUIDS.  323 

and  this  would   make  z'  too   great.     Taking  a  mean  between  z^  and 
z'  as  the  true    value,  we  find 

z  =  ^^^t_L  ^  [1  4-  ^  (/   4-  /'  _  64°) .  0,001>08]  A". 
Replacing  X  by  its  value, 

^~~~I),  1-0,002551  cos  2  vj.     ^  ^"L/i  ^  1 +  (r- r)0,0001001  J 

The  factor  ~ — '-^t  we  have  seen,  is  constant,  and  it  only  re- 
mains to  determine  its  value.  For  this  purpose,  measure  with 
accuracy  the  difference  of  level  between  two  stations,  one  at  the 
base  and  the  other  on  the  summit  of  some  lofty  mountain,  by 
means  of  a  Theodolite,  or  levelling  instrument — this  will  give  the 
value  of  z  ;  observe  the  barometric  column  at  both  stations — this 
will  give  h  and  h^  ;  take  also  the  temperature  of  the  mercury  at 
the  two  stations — this  will  give  T  and  T' ;  and  by  a  detached 
thermometer  in  the  shade,  at  both  stations,  find  the  values  of 
t^  and  t'.  These,  and  the  latitude  of  the  place,  being  substituted  in 
the  formula,  every  thing  will  be  known  except  the  co-efficient  in 
question,  which  may,  therefore,  be  found  by  the  solution  of  a  simple 
equation.     In   this   way,  it   is   found   that 

^^^^  ^'"    =  60345,51  English  feet ; 

v/hlch  will  finally  give  for  z,  « 

/^       i_f.i(/.|-<'_G4°)0.00208  r/', 1 1 

z=6034o,51.     1  _  0002551  cos2Y     ^  ^^Lr^l-f(r-r)0,000100lJ 

To  find  the  difiercnce  of  level  between  any  two  stations,  the  lati- 
tude of  the  locality  must  be  known  ;  it  will  then  only  be  necessary 
to  note  the  barometric  columns,  the  temperature  of  the  mercury, 
and  that  of  the  air  at  the  two  stations,  and  to  substitute  these 
observed    elements    in    this    formula. 

Much  labor  is,  however,  saved  by  the  use  of  a  table  for  the 
computation  of  these  results,  and  we  now  proceed  to  explain  how  it 
may  be  formed  and  used. 


524  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Make 

60345,51  [1  +  (^;  +  t'  -  64)  0,00104]  =  A, 


Then  will 


1  —  0,002551  cos  2  4^ 

_1 

1  +  {T  -  T)  0,0001 

r .  h 


=  C. 


h 
z  =  AB'  [log  C  +  log  h,  -  log  h]  ; 

and   taking   the   logarithms  of  both   members, 

log  z  =  log  ^  4-  log  ^  +  log  [log  C  +  log  h,  -  log  K]  ■  .  (464) 

Making  t^  +  t'  to  vary  from  40°  to  162°,  which  will  be  sufficient 
for  all  practical  purposes,  the  logarithms  of  the  corresponding  values 
of  A  are  entered  in  a  column,  under  the  head  A,  opposite  the 
values  t,  +  t',  as    an   argument. 

Causing  the  latitude  4.  to  vary  from  0°  to  90°,  the  logarithms 
of  the  corresponding  values  of  B  are  entered  in  a  column  headed 
B,  opposite   the   values   of  4/. 

The  value  of  T  —  T  being  made,  in  like  manner,  to  vary 
from  —  30°  to  +  30°,  the  logarithms  of  the  correspojiding  values 
of  C  are  entered  under  the  head  of  C,  and  opposite  the  values  of 
T  —  T.  In  this  way  a  table  is  easily  constructed.  Table  IV  was 
computed  by  Samuel  Howlet,  Esq.,  from  the  formula  of  Mr.  Francis 
Daily,  which  is  very  nearly  the  same  as  that  just  described,  there 
being   but   a    trifling  diflference   in    the   co-efficients. 

Taking  Equation  (464)  in  connection  with  Table  IV,  we  have  this 
rule  for   finding  the    altitude  of  one   station   above   another,  viz.  :— 

Take  the  logarithm  of  the  barometric  reading  at  the  lower  station, 
io  which  add  the  number  in  the  column  headed  C,  opposite  the  ob- 
served value  of  T  -  r,  and  subtract  from  this  sum  the  logarithm 
of  the  barometric  reading  at  the  upper  station;  take  the  logarithm 
of  this  difference,  to  ivhich  add  the  numbers  in  the  columns  headed 
A  and  B,  corresponding  to  the  observed  values  of  t^  +  t'  and  ■],  ; 
the  sum  will  be    the    logarithm   of  the  height   in   English  feet. 


MECHANICS    OF    FLUIDS.  325 

Example. — At  the  mountain  of  Guanaxuato,  in  Mexico,  M.  Hum- 
boldt observed   at   the 

Upper  Station.  lAitrer  Station. 

Detached   thermometer,    t'  =  70°,4 ;      t^   z=  T7°,6. 
Attached  "  r  =  70,4 ;       T  =  77,6. 

Barometric  column,  k   =  23, GO  ;     h^  =  30,05. 

What  was   the  difference  of  level  1 
Here 

t^  -{-  t'  =  148°  ;     T  —  T  —  7°,2  ;     Latitude  21°. 

To   log     30,05  =  1,4778445 
Add  Cfor  7°,2  =  0,0003165 
1,4781610 
Sub.  log  23,66  =  1,3740147 
Log  of     -     -     -     0,1041463  =  -  1,0176439 
Add  A  for  148°  -     ...     -        4,8193975 
Add  ^  for  2P    -     -     -     -     =        0,0008689 
6885J1 3,8379103; 

whence   the   mountain   is   6885,1  feet  high. 

It  will  be  remembered  that  the  final  Equation  (464)  was  deduced 
on  the  supposition  that  the  air  is  in  equilibrio — that  is  to  say, 
when  there  is  no  wind.  The  barometer  can,  therefore,  only  be  used 
for  levelling  purposes  in  calm  weather.  Moreover,  to  insure  accu- 
racy, the  observations  at  the  two  stations  whose  difference  of  level 
is  to  be  found,  should  be  made  simultaneously,  else  the  temperature 
of  the  air  may  change  during  the  interval  between  them  ;  but  with 
a  single  instrument  this  is  impracticable,  and  we  proceed  thus,  viz. : 
Take  the  barometric  column,  the  reading  of  the  attached  and  detached 
thermometers,  and  time  of  day  at  one  of  the  stations,  say  the 
lower;  then  proceed  to  the  upper  station,  and  take  the  same 
elements  there ;  and  at  an  equal  interval  of  time  afterward,  observe 
these  elements  at  the  lower  station  again ;  reduce  the  mercurial 
columns  at  the  lower  station  to  the  same  temperature  by  Equation 
(394),  take  a  mean  of  these  columns,  and  a  mean  of  the  tempera- 
tures  of    the   air   at  this   station,  and   use   these   means   as   a   single 


326  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

set    of  observations    made    simultaneously    with   those    at    the   higher 
station. 

Example. The   following   ohservations  were    made    to    determine 

the  height   of  a  hill   near  West  Point,  N.  Y. 

Vpver  Station.  Lower  Station. 

(1)  (2) 

Detached  thermometer,  t'   =  57°  ;       t^   =  56°      and  61°. 
Attached  "  r  =  57,5  ;     T  =  56,5     and  63. 

Barometric  column,  h   =  28,94:  ■  h,  =  29"62  and  29"63. 

First,  to  reduce  29,63  inches  at  63°,  to  what  it  w^ould  have 
been    at  56°,5.     For    this   purpose,  Equation    (394)  gives 

in. 

J,  (1  +  T  -  Tx  0,0001)  =  29,63  (1  -  6,5  x  0,0001)  ==  29,611. 

Then 

^^,t)^  +  ^v^,oii  ^  29,6105. 
'  2 

,  =  5?!  +  ^   .  .  =  58",6, 

t,  +  t'  =  5S°,5  +57°-  -  =  115°,5, 
T—  T  =  56°,5  -  57°,5  -  =  —  1°. 

To  log  29,6105   =  1,4714458 

Add  C  for  —  1°  =  9,9999566 

1,4714024 

in. 

Sub.  log  of  28,94  =  1,4614985 

Log  of  -    -    -    -    0,0099039  =  -  3,9958062 

Add  A  for  115°,5      -     -     -     =  4,8048112 

Add  JB  for  41°,4   -    -    -    -     =  0,0001465 

632,07 2,8007639; 

whence  the  height   of  the  hill  is  632,07  English  feet. 

MOTION   OF  HEAVY   INCOMPRESSIBLE   TXUIDS   IN   VESSELS. 

1 285. — A  heavy  homogeneous  liquid  moving  in  a  vessel,  may 
be  regarded  as  an  assemblage  of  indefinitely  thin  strata  arranged 
perpendicularly  to  the  direction  of  the  motion,  and  these  strata  may 


MECHANICS     OF    FLUIDS. 


327 


be  regarded  as  so  many  solid  bodies,  provided  we  attribute  to 
them  the  property  of  contracting  ahd  expanding  in  different  direc- 
tions so  as  to  maintain  a  constant  volume  in  adapting  themselves 
to  the  varying  cross  section  of  the  vessel  in  which  they  are  movin<T, 

Let  A  B  C  D  he  a  vessel  of 
which  the  axis  is  vertical,  and 
whose  horizontal  sections  vary 
only  by  insensible  degrees  ;  sup- 
pose the  fluid  divided  into  an 
indefinite  number  of  thin  level 
strata  whose  volumes  are  equal 
to  one  another.  We  may  sup- 
pose that  at  the  end  of  each 
element  of  time  any  one  stratum 
occupies  the  space  filled  by  the 
stratum  Avhich  preceded  it  at 
the  commencement  of  this  cle- 
ment. 

The  horizontal  velocities  of  the  particles  of  the  fluid  may  be 
disregarded,  and  the  vertical  velocity  of  any  one  of  them  will  be 
the  same  as  that  of  every  other  particle  in  the  same  stratum. 
The  motion  of  the  fluid  will  be  known  when  we  know  that  of  any 
one   stratum. 

§286. — Taking  the  axis  of  z  vertical  and  positive  upwards,  we 
shall  have,  in  Equations  (400)  and  (401), 


A"=0;      F=0;     Z  =  -  ff;     u  =  0;     v  =  0, 


d  io 


and,  therefore, 


I)     dx  '     D     d>j~     ' 

1      dp  dw 

'D'Tz~~^'^'dt' 


in  which  it  will  be  recollected  that  w  is  the  velocity  of  any  one 
particle,  and  therefore  of  the  stratum  to  which  it  belongs,  in  the 
direction  of  z. 


828         ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Multiplying  the   last    equation  by  Ddz,  and  integrating,  we  have 
p^  -  D.g.z^  D-f~-dz+  C.    .     .     .     (465) 

Take   the  followhig  notation,  viz. : — 

s    =  the  variable  area  of  the    stratum  whose  velocity  is  w. 

s    =  the  constant  area  of  any  determinate   horizontal  section  of  the 

vessel,  as  CD. 
S  =  the  area  of  the   section  of  the    vessel   by  the  upper  surface  of 

the   liquid ;  this   may  be  constant  or  variable,  according  as  the 

upper  surface  is  stationary  or  movable. 
w   =  velocity  of  the   stratum   passing   the   section   s^  at  CD,  at  the 

time  t. 
The    fluid    being    incompressible,    the    same    volume    must    pass 
everv  horizontal   section  in  the   same  interval  of  time ;  and  hence 


or 


and 


but 


IV-S,   =  W-  s, 


s 


dw   _    s,     dw,  _  If    ^    il. 

~dT  -~I'^A         '"'''    dz'  dt'  s^' 


dz  w.s. 

w  = 


dt  s 

Substituting    this   in   the   last   term,    and   multiplying   by   dz,   we 


have 


dw         ■  dw,     dzds 


and   integrating,  regarding  z,  and   therefore  s,  as    variujle, 

J    dt  '     dt  J    s  2       &^ 

which,  in  Equation  (465),  gives 

dxi).    rdz         ^w^     s,2         ^       (Aarf\ 
P  =  -Dgz  +  D.s,-  -jfj-  -  i)-  .  ^  +  C  .  (467) 


MECHANICS     OF     FLUIDS.  329 


To  find  the  value  of  C,  let  p  -  P , ,    ^vhen    z  =  z,,  which    corre- 
sponds to    the  section   CD  of  the    liquid  ;  then  will 

dw,      r       d . 

z-z, 

Avhich,    subtracted  from    the  equation    above,  gives 


dw,      r       dz         ^   w^      s  2 


V 


P,=-^,,-,)+i>-«/^/;^-^-2'¥[^- ']•(««) 


Also,  if  P'  denote  the  pressure  at  the  upper  surface  corresponding 
to  -syhich  z  —  z',  we   have 

Ivfow  z'  —  z,  =  h  =  height  of  the  fluid  surface  above  the  section 
C£>;   whence,  by  substitution  and   transposition, 

The  quantity  of  fluid  flowing  through  every  section  in  the  same 
time   being   equal,  we   also  have 

-  Sdh  z^  s,.w^.dL (471) 

By  means  of  this   equation,  t   may   be    eliminated  from    Equation 

(470)  ;    then    knowing  the   quantity  of  the  liquid,   the   size  and  figure 

f^'dz         r^dz 
of  the   vessel,  we  will   know  h,  S,  and   the  integral    /      —  =   /     — , 

in  which  s  is   a    function  of  z. 

gog7^ — Xhe  value  of    -j-^   being  found  from  Equation  (470),  and 

substituted  in  Equation  (46S),  this  latter  equation  will  give  the  value 
of  the  pressure  p  at  any  point  of  the  fluid  mass  as  soon  as  u\  be- 
comes  known. 

Two  cases  may  arise.  Either  the  vessel  may  be  kept  constantly 
full  while  the  liquid  is  flowing  out  at  the  bottom,  or  it  may  be 
suflfered   to   empty  itself. 

§288. — To  discuss  the  case  in  which  the  vessel  is  always  full,  or 
the   fluid   retains  the  same  level  by  being  supplied  at  the  top  as  fust 


330  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

as   it  flows  out  at  the  bottom,  the   quantity  h  must   be  constant,  and 
Equation  (471)  will    not   be   used. 
And  making,  in  Equation  (470), 

s 

P'  -P. 


y'^''  dz 


and   solving  with  respect   to  d  t,  we   have 

'f  =  ^% («^) 

Now  three  cases  may  occur. 

1st.  S  may  be  less   than  s, ,  and   C  will  be  positive. 

2d.    S  may  be  equal    to  s^ ,   in  which    case   C  will   be  zero. 

3d.    S  may  be  greater  than  s^ ,  when   C  will  be  negative,  and  this 

is   usually  the  case  in  practice. 
In  the  first  case,  when  C  is  positive,  we  have,  by  integrating  Equa- 
tion (472),  and  supposing   t  =  0,  when  w^  —  0, 

t=-^-tan-'\o,J^', (473) 


whence, 

-•tan   ^    ^^     't. (4i4) 

from  which  we  see  that  the  velocity  of  egress  increases  rapidly  with 
the   time :    it   becomes    infinite  when 


■JBC  * 

or 

t=^J!l2A^ (475) 

^^BO 

When  (7  =  0,  then  will  the   integration  of  Equation  (472)  give 


MECHANICS    OF     FLUIDS.  831 

or  replacing  A  and  B  by  their  values?,  and  finding  the  value  of  w^ , 

«'.  =  — ^^.^7 '' (^''^ 


/'» a  z 


whence,  the  velocity  varies  directly  as  the  time,  as  it  should,  since 
the  whole  fluid  mass  would  fall  like  a  solid  body  under  the  action 
of  its  own  weight. 

When   C  is   negative,  the    intcgniticfn  gives 


t  =  ==  •  log 


2  -vAFa        °     VB  -  -/^'   ' 


whence, 


e     -^        -1         /B 

w.  = 


'iy/~BC    ^  V     C   ' 


(478) 


e    -'        +1 
in  which  e  is  the  base  of  the  Naperian  system  of  logarithjns  =  2,718282. 
If  the   section    >S   exceeds   s^  considerably,  the   exponent    of  e  will 
soon   become  very  great,  and  unity  may  be  neglected  in  comparison 
with    the   corresponding    power  of  e  ;  whence, 


^-0-^^) 


1  -^ 

52 


(479) 


that  is  to   say,  the  velocity  will   soon  become  constant. 

If  the  pressure  at  the  upper  surface  be  equal  to  that  at  the  place 
of  egress,  which  would  be  sensibly  the  case  in  the  atmosphere, 
P'  -  P^  =  0,  and 

2  q  h 


1  _^; (480) 

and  if  the  opening  below  become  a  mere  orifice,  the  fraction 

?  2 

1^  =  0; 

and 

w^  =  V^Jh; (481) 


332  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

that  is  to  say,  the  velocity  with  Avhich  a  heavy  liquid  will  issue 
from  a  small  orifice  in  the  bottom  of  a  vessel,  when  subjected  to 
the  pressure  of  the  superincumbent  mass,  is  equal  to  that  acquired 
by  a  heavy  body  in  falling  through  a  height  equal  to  the  depth  of 
the  orifice  below  the  upper  surface  of  the  liquid.  The  velocities 
given  by  Equations  (479),  (480),  (481),  are  independent  of  the 
figure  of  the  vessel. 

If  the  velocity  w^  be  multiplied  by  the  area  s^  of  the  orifice,  the 
product  will  be  the  quantity  of  fluid  discharged  in  a  unit  of  time. 
This  is  called  the  expense.  The  expense  multiplied  by  the  time  of 
flow  will   give  the  whole  quantity  discharged. 

R289. The   velocity  w^  being   constant  in  the  case   referred  to  in 

Equation  (479),  we   shall  have 

-If-'' 

and  Equation  (468)  becomes 

or,   substituting  the  value  of  %v^,  given   by  Equation  (470), 

p  =  P,-Dg  {z  -  z')  +  {Dgh  +  P'  -  P,)  ■"-, ;    .     .     (482) 

whence,  it  appears,  that  when  the  flow  has  become  uniform,  the  pres 
sure  upon  any  stratum  is  wholly  independent  of  the  figure  of  the 
vessel,  and  depends  only  upon  the  area  s  of  the  stratum,  its  distance 

s  2 
from  the  upper   surface  of   the  fluid,  and  upon  the  ratio  ^. 

s  290.— If  the  vessel  be  not  replenished,  but  be  allowed  to  empty 

O 

itself,   h  will    be   variable,    as   will   also   S   except  in   the   particular 
cases  of  the  prism  and  cylinder. 
Making 

w,  =  -v/27^ (483) 


MECHANICS     OF     FLUIDS.  333 

in  which  H  deaotes  the  height  due  to  the  velocity  of  discharge :  we 
have 

and,  Equation  (-471), 
and  by  integration. 


dt  = ^= ; (485) 


,.C__L^./^ (486) 

To  effect  the  integration,  S  and  H  must  be  found  in  terms  of  h. 
The  relation  between  S  and  h  will  be  given  by  the  figure  of  the 
vessel.  Then  to  find  the  relation  between  H  and  h,  eliminate  w,, 
d  u\ ,  and  d  i  from  Equation  (470),  by  the  values  above,  and  we  have 

or,  dividing  by 

-(^+^)  -O-g) 

—\~r- dh  +  dll .-y - 

r>'  dz  ^   f'^  dz 

'Jog  'Jos 

and  making 

S,2   /        -  */-  /        - 

'Jos  'Jos 

Qdh  +  dll  +  RHdh  =  0 (488) 

fRdh 

Multiplying  by  e        , 

fRdh  flidh  fjldh 

dh'  Q  '  e         -\-  dH  '  c         +  JI  .e  x  R  d  h  =  0  ; 


^-— dh  +  dll .-^  .H-dk=0-  (4S7) 

'^  dz  „    f'*  dz 


334  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

or 

fRdh  ,        fRdh 

dh'  Q  '  e         +  d  (He         )  =  0; 

/fRdh  fRdh 

dh'  Q-e         +  He         =  C;  .     .     .     .     (489) 

-fRdh  fRdh     ^   ^     ^     ^     (490) 

H=e        -    (^C  -   fdkQ-e        \  ^       ^ 


and  integrating 


whence, 


The  constant  must  result  from  the  condition,  that  when  H  =  0, 
k  must  be  h^,  the  initial  height  of  the  fluid  in  the  vessel. 

Thus  H  becomes  known  in  terms  of  h,  and  its  value  substituted 
in  Equation  (486)  will  make  known  the  time  required  for  the  fluid 
to  reach  any  altitude  h.  The  constant  in  Equation  (486)  must  be 
determined,  so   that  when   ^  =  0,   h  =  h^. 

1 291. — The  mode  of  solution  here  indicated  is  direct  and  general; 
but  analysis,  in  its  application  to  the  motion  of  fluids,  often  pre- 
sents itself  under  forms  which  require  us,  in  particular  cases,  to 
adapt  the  mode  of  solution  to  the  peculiarities  of  each  special  case. 
Take,  for  example,  the  case  of  a  right  cylinder  or  prism.  Here  S 
will  be  constant,  and  equal  to  s. 

dz         h 
s  S 


/: 


Moreover,  let  us  suppose  P'  —  P^  =  0,  which  would  be  sensibly 
true  were  the  fluid  to  flow  into  the  atmosphere  that  rests  upon  its 
upper  surfece.  Also,  for  the  sake  of  abbreviation,  make  —  =  k, 
then  will 


R  -  - 


h     ~     h    ' 

e=F; 

and  Equation  (488)  becomes 

k'' .h.dh  +  h.dH  +  {\  -k'').H.dh=0.     .     (491) 


MECHANICS    OF    FLUIDS,  335 


—  fc- 
Multiplying  by  It       ,  we  have 

/t2 .  ii'^^dh  -f  h~^\iH  +  (1  -  r-)  ?r^  dh-  IT  =  0, 


or 


2  -  A;2 
and  by  integration, 


p  2  -  fc2  1  _  ^2 

dh         +  d  {h         X  //)  =  0; 


Z-2  2  -  fc2  1  _  A2 


2    -    ^2 

Now,  Avhen  h  z=  k^ ,  then  will  //  =  0  ;    whence, 


F  2  _  fc2 


2  -  ^: 
and 


F  2-k'^  l-fc2  p  2-^2 


2  -  A;2  -  2  -  ^2       '         ' 

whence, 

~   2  -  Ar2  *  1  _  ^2  ' 

/i 

multiplying  both  numerator  and  denominator  by  h, 

^=^^'['-(^)'"]-  •  •  •  («^) 

which    substituted   in   Equation  (486),  gives 

^           /k~  —  2      /^  dh 

t  =  C  -X    -— .  f  =,      .     .     (493) 

in  which   the   only  variable  is  /(. 

§292. — The   particular  case   in   which  P  _  o,  gives    to  this  value 

for  t  the  form  of  indetormination.     When   this   occurs,  we  must  have 

recourse    to    the    form  assumed  by  Equation  (491),  which,  under  this 
supposition,  becomes 

2  h  d  h  +  Itdll  —  Hdh  ^  0  ; 


336  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

multiplying   by  h    ", 

2h~^dh  +  h~\dH  -H.h'^dh  =  0, 

2  log  A  +  y  =  C ; 

and  because  H  =  0  when  h  =  h^ , 

2  log  A,  =  C; 
whence, 

•  ^  =  2  /i  •  log  -^, 

and  this,  in  Equation  (486),  gives 

dh 


t  =  C 


Lf 


Making  -^  =  — ?  this  becomes 


t 


loj: 
'     X 


The   value  of   C  is   determined   by   making  a;  =  1  when  t  ■=  0. 

1 293. — If  the  orifice  be  very  small  in  comparison  with  a  cross 
section  of  the  prismatic  or  cylindrical  vessel,  then  will  H  —  h^  and 
Equation  (486)  gives 

Making   t  =  0   when  A  =  7i, ,  we   have 

i=-^-(VA;-/A), (494) 

and  for  the  time  required  for  the  vessel  to  empty  itself,  A  =  0,  and 


MECHANICS     OF     FLUIDS.  337 

Now,  with    the    same   relation  of  the    orifice    to    the   cross   section 
of  the  cylindrical  vessel,  we   have,  Equation  (483), 

and   for   the    quantity    of    fluid    discharged    in   the    time    /,    M-hen    the 
vessel    is    kept  full, 

and    if  this   be   equal    to  the   contents   of  the   vessel, 


whence, 

t  =  A     .  [K.     '* 

That  is.  Equation  (495),  the  time  required  for  a  prismatic  or  cylin- 
di'ical  vessel  to  discharge  itself  through  a  small  orifice  at  the 
bottom  is  double  that  required  to  discharge  an  equal  volume,  if 
the  vessel  were   kept  full. 

§  294. — The     orifice    being    still    small,    we    obtain,    from    Equa- 
tion (485), 

—  =  -^  .  /27r; 

whence  it  appears  that,  for  a  cylindrical  or  prismatic  vessel,  the 
motion  of  the  upper  surface  of  the  fluid  is  uniformly  retarded.  It 
will  be  easy  to  cause  S  so  to  vary,  in  other  words,  to  give  the 
vessel  such  figure  as  to  cause  the  motion  of  the  upper  surface  to 
follow  any  law.  If,  for  example,  it  were  required  to  give  such  figure 
as  to  cause  the  motion  of  the  upper  surface  to  be  uniform,  then 
would  the  first  member  of  the  above  equation  be  constant ;  and, 
denoting   the  rate  of  motion   by  a,  we   should   have 


whence, 

o  2     O  fj  h 
C  6,     .  ^  1/  II' 

^^  =  ^^~  ■' 

but   supposing   the    horizontal    sections    circular, 

S^  :=  -rr-  r*  =^  — > 


338  ELEMENTS     OF    ANALYTICAL    MECHANICS, 

and,  therefore, 


whence  the  radii  of  the  sections  must  vary  as  the  fourth  root  of 
their  distances  from  the  bottom.  These  considerations  apply  to  the 
construction  of  Clepsydras  or   Water   Clocks. 


MOTION   OF   ELASTIC   FLUIDS   IN   VESSELS. 

8  295. — As  in  the  case  of  incompressible,  so  also  in  that  of 
elastic  fluids,  it  is  assumed  that  in  their  movement  through  vessels, 
they  arrange  themselves  into  parallel  strata  at  right  angles  to  the 
direction  of  the  motion.  The  quantity  of  matter  in  each  stratum 
is  supposed  to  remain  the  same,  while  its  density,  which  is  always 
uniform  throughout,  may  vary  from  one  position  of  the  stratum  to 
another  ;    hence,  the   volume   of  each    stratum  may  vary. 

All  lateral  velocity  among  the  particles  will  be  supposed  zero ; 
and  as  the  weight  of  the  elements  of  elastic  fluids  is  insignificant 
in  comparison  to  their  elasticity,  the  former  will  be  disregarded. 
The  motion  will,  therefore,  be  due  only  to  the  elastic  force  arising 
from  some  force  of  compression ;  and  as  the  fluid  will  be  supposed 
to  communicate  freely  with  the  air,  or  with  a  vessel  partly  filled 
with  some  other  elastic  fluid,  this  force  within  may  be  greater  or  less 
than   it  is   on  the   exterior  of  the   vessel. 

R296. Assuming    the    axis    of   the   vessel    horizontal,    take    that 

line  as  the  axis  of  x. 
Then,  by  the  supposi- 
tion above,  will 

X  =  0 
Y  =0 
Z  =0 
V  =  0 
w  =  0;  -Q  2i' 


Z 

A         ^ 

i 

j   — -v^ 

D 

I    — -^ 

/      ^ 

MECHANICS     OP^    FLUIDS.  839 

and  Equations  (400)  give 

1   .4^  =  _(4ji)_4«.„.   ....     (49G) 
D      dx  \dt/         dx  ^       ' 

Moreover,  if  we  suppose  the  motion  to  have  been  established 
and  become  permanent,  the  velocity  of  a  stratum  as  it  passes  any 
particular  cross  section  of  the  vessel  will  always  be  constant,  and 
the  quantity  of  fluid  which  flows  through  every  cross  section  will 
be  the  same.  Hence  the  partial  differential  of  u  in  regard  to  the 
time,  that  is,  supposing  r,  y,  2,  to  be  constant,  must  be  zero,  and 
the   above   equation  reduces   to 

d  p  =^  —  D  .  u  .  d  u. 
.From   Mariotte's   law.  Equation  (389), 

p  =  P.I>, 
and   by   division, 


dp  _  _    l^ 
y  ~  ~  F 


u  d  u, 


and   by  integration, 


logp=  C-~u\    ......     (407) 


To  determine  the  constant,  let  p,  be  the  pressure  at  the  opening 
CD,  that  is,  the  pressure  of  the  atmosphere,  and  denote  by  u^  the 
velocity  of  the    fluid   at   this   point,  then  will 

1 


and  by   subtraction, 


logp.  =  C      .^^ 


log^  =  2^-(".^-«^). (498) 


Denote  by  s  the  area  of  any  section  of  the  vessel  A'  B\  at  which 
the  pressure  is  p)  and  velocity  u,  byZ>  the  density  of  the  fluid  at 
this  section,  and  by  D^  that  at  the  section  CD  equal  to  s, .  Then, 
since  the  quantities  of  fluid  flowing  through  these  sections  in  a  unit 
of  time   must  be   equal,  we   have 

D  .  s  .u  —  D, .  s,  .  ?r  ; 


340         ELEMENTS     OF    ANALYTICAL    MECHANICS, 
but,  §244, 

whence, 


D 

= 

V 

s .  u 

= 

p,^, 

^'n 

n  — 

h 

s,n, 

-> 

2)  .s 
which,  in  Equation  (498),  gives 

If  p'  denote   the   pressure  exerted   by  the   piston  A  B,  and  S   de- 
note its  area,  we   have 


'-4=^i-'-(n^^-'  ■  ■ 


whence. 


'2FAos 


P, 


(500) 


(501) 


This    is    the   velocity    with    which    the    fluid    will     issue    into    the 
atmosphere  or  other  fluid  whose  pressure  on  the  unit  of  surfaces  is  p, . 

§  297. — The  volume   discharged  in  a  unit  of   time  is 


1-2  P  ■  l0£ 


V,  s,  =  s, 


P, 


while    under   the   pressure  p^ ;    and    under   a   pressure   equal    to    that 
on    the    unit    of  surface   of  the    piston, 
or  top  of  a  gasometer,  and  which  would 
be  indicated  by  a  gauge,   since  the  vol- 
umes are  inversely  as  the  pressures. 


.,...  =  ^. 


(502) 


MECHANICS    OF    FLUIDS.  341 

298. — Dividing  Eipatiou  (499)  by  Equation  (500),  we  have 


P,  \P 

whicli  will  give   the  pressure  ^y  at  any  section  of    the  vessel. 

§299. — If  the  opening   CD  is  very  small  in  reference  to  ^  ^,  the 
velocity  u,  will   become,  Equation    (501), 


^.  =  \l'^  ^  •  log  ^  ; (504) 

and    the  volume   of   fluid  discharged  in  a  unit  of  time  and  of  a  den- 
sity equal  to  that  pressing  upon  the  gauge. 


\/2P.log|^; (505) 


and  Equation  (503)  becomes 

log^ 
"^  P, 


=  1 


los 


p_ 
p, 


\p  s  ) 


§  300. — A  stream  flowing  through  an  orifice  is  called  a  vein.  In 
estimating  the  quantity  of  fluid  discharged,  it  is  supposed  that  there 
are  neither  within  nor  without  the  vessel  any  causes  to  obstruct  the 
free  and  continuous  flow ;  that  the  fluid  has  no  viscosity,  and  does 
not  adhere  to  the  sides  of  the  vessel  and  orifice  ;  that  the  particles 
of  the  fluid  reach  the  upper  surface  with  a  common  velocity,  and  also 
leave  the  orifice  with  equal  and  parallel  velocities.  None  of  these 
conditions  are  fulfilled  in  practice,  and  the  theoretical  discharge  must, 
therefore,  differ  from  the  actual.  Experience  teaches  that  the  former 
alwavs  exceeds  the  latter.  If  we  take  water,  for  example,  which  is 
far  the  most  important  of  the  liquids  in  a  practical  point  of  view, 
we  shall  find  it  to  a  certain  degree  viscous,  and  always  exhibiting  a 
tendency  to  adhere  to  ununctuous  surfaces  with  which  it  may  be 
brought    in    contact.       When    water    flows    through    an    opening,   the 


342         ELEMENTS     OF    ANALYTICAL    MECHANICS. 

adhesion  of  its  particles  to  the  surface  will  check  their  motion,  and 
the  viscosity  of  the  fluid  will  transmit  this  effect  towards  the  interior 
of  the  vein;  the  velocity  will,  therefore,  be  greatest  at  the  axis  of 
the  latter,  and  least  on  and  near  its  surface  ;  the  inner  particles  thus 
flowing  away  from  those  without,  the  vein  will  increase  in  length  and 
diminish, in  thickness,  till,  at  a  certain  distance  from  the  orifice,  the 
velocity  becomes  the  same  throughout  the  same  cross-section,  which 
usually  takes  place  at  a  short  distance  from  the  aperture.  This 
effect  will  be  increased  by  the  crowding  of  the  particles,  arising  from 
the  convergence  of  the  paths  along  which  they  approach  the  aper- 
ture, every  particle,  which  enters  near  the  edge,  tending  to  pass 
obliquely  across  to  the  opposite  side.  This  diminution  of  the  fluid 
vein  is  called  the  veinal  contraction.  The  quantity  of  fluid  discharged 
must  depend  upon  the  degree  of  veinal  contraction,  and  the  velocity 
of  the  particles  at  the  section  of  greatest  diminution ;  and  any  cause 
that  will  diminish  the  viscosity  and  cohesion,  and  draw  the  particles 
in  the  direction  of  the  axis  of  the  vein  as  they  enter  the  aperture, 
will  increase  the  discharge. 

Experience  shows  that  the  greatest  contraction  takes  place  at  a 
distance  from  the  vessel  varying  from  a  half  to  once  the  greatest 
dimension  of  the  aperture,  and  that  the  amount  of  contraction  de- 
pends  somewhat  upon  the  shape  of  the  vessel  about  the  orifice 
and  the  head  of  fluid.  It  is  further  found  by  experiment,  that  if  a 
tube  of  the  same  shape  and  size  as  the  vein,  from  the  side  of  the 
vessel  to  the  pl^ace  of  greatest  contraction,  be  inserted  into  the 
aperture,  the  actual  discharge  of  fluid  may  be  accurately  computed 
by  Equation  (502),  provided  the  smaller  base  of  the  tube  be  sub- 
stituted for  the  area  of  the  aperture;  and  that,  generally,  without 
the  use  of  the  tube,  the  actual  may  be  deduced  from  the  theoretical 
discharge,  as  given  by  that  equation,  by  simply  multiplying  the 
theoretical  discharge  into  a  co-efficient  whose  numerical  value  depends 
upon  the  size  of  the  aperture  and  head  of  the  fluid.  Moreover, 
all  other  circumstances  being  the  same,  it  is  ascertained  that  this 
co-eflScient  remains  constant,  whether  the  aperture  be  circular,  square, 
or  oblong,  which  embrace  all  cases  of  practice,  •  provided  that  in 
comparing  rectangular  with  circular  orifices,  we  compare  the  smallest 


MECHANICS     OF     FLUIDS. 


343 


dimension  of  the  former  whh  the.  diameter  of  the  latter.     The  value 

of  this   co-efficient   depends,   therefore,  when    other   circumstances   are 

the   same,    upon   the    smallest   dimension   of    the   rectangular    orifice, 

and    upon  the  diameter  of  the   circle,  in  the  case  of  circular  orifices. 

But   should   other   circumstances,  such   as   the   head  of  fluid,  and  the 

place  of  the   orifice,  in   respect   to  the  sides 

and   bottom    of  the    vessel,    vary,   then    will 

the   co-efficient   also  vary.      When   the   flow 

takes   place    through  thin  plates,  or   through 

orifices  whose   lips    are    bevelled    externally, 

the  co-efficient  corresponding  to  given  heads 

and    orifices,    may   be   found    in    Table    V, 

provided    the    orifices    be    remote   from    the 

lateral    faces    of   the   vessel.      This   table   is 

deduced    from    the    experiments    of   Captain 

Lesbros,  of  the  French  engineers,  and  agrees 

with   the   previous .  experiments    of  Bos^ut,  Michelotti,  and  others. 

As  the  orifice  approaches  one  of  the 
lateral  fiices  of  the  reservoir,  the  contrac- 
tion on  that  side  becomes  less  and  less, 
and  will  ultimately  become  nothing,  and  the 
CO- efficient  will  be  greater  than  those  of  the 
table.  If  the  orifice  be  near  two  of  these 
faces,  the  contraction  becomes  nothing  on 
two  sides,  and  the  co-efficient  will  be  still 
greater. 

Under  these  circumstances,  we  have  the 
following  rules  : — Denote  by  C  the  tabular, 
and  by  C"  the  true  co-etficient  corresponding 
to  a  given  aperture  and  head  ;  then,  if  the 
contraction  be   nothing  on   one   side,  will 

C  =  1,03  C; 
if  nothing   on   two   sides, 

C  =  1,00  C; 
if  nothing    on   three    sides, 

C  =  1,12C; 


\;Vi 


344  ELEMENTS     OF     ANALYTICAL     MECHANICS. 

and  it  must  be  borne  in  mind,  that  these  results  and  those  of  the 
table  are  applicable  only  Avhen  the  fluid  issues  through  holes  in 
thin  plates,  or  through  apertures  so  bevelled  externally  that  the 
particles  may  not  be  drawn  aside  by  molecular  action  along  their 
tubular   contour. 

g301. — When  the  discharge  is  through  thick  plates  without  bevel, 
or  through  cylindrical  tubes  whose  lengths  are  from  two  to  three 
times  the  smaller  dimension  of  the  orifice,  the  expense  is  increased, 
the  mean  coefficient,  in  such  cases,  augmenting,  according  to  experi- 
ment, to  about  0,815  for  orifices  of  which  the  smaller  dimension 
varies  from  0,33  to  0,66  of  a  foot,  under  heads  which  give  a  coeffi- 
cient 0,619  in  the  case  of  thin  plates.  The  cause  of  this  increase  is 
obvious.  It  is  within  the  observation  of  every  one,  that  water  wall 
wet  most  surfaces  not  highly  polished  or  covered  with  an  unctuous 
coating — in  other  words,  that  there  exists  between  the  particles  of 
the  fluid  and  those  of  solids  an  affinity  which  will  cause  the  former 
to  spread  themselves  over  the  latter  and  adhere  with  considerable 
pertinacity.  This  affinity  becoming  eflective  between  the  inner  sur- 
face of  the  tube  and  those  particles  of  the  fluid  which  enter  the 
orifice  near  its  edge,  the  latter  will  not  only  be  drawn  aside  from 
their  converging  directions,  but  will  take  with  them,  by  the  force  of 
viscosity,  the  other  particles,  with  which  they  are  in  sensible  contact. 
The  fluid  filaments  leading  through  the  tube  will,  therefore,  be  more 
nearly  parallel  than  in  the  case  of  orifices  through  thin  plates,  the  con- 
traction of  the  vein  will  be  less,  and  the  discharge  consequently 
greater. 


PART    III. 


APPLICATIOX    OF    THE    PRECEDING    PRINCIPLES    TO 
SIMPLE    MACHINES,    PUMPS,    ETC. 

§  302. — Any  device  by  which  the  action  of  a  force  may  be  received 
at  one  place  and  transmitted  to  another  is  called  a  Machine. 

There  are  usually  seven  elementary  machines  discussed  in  ^fe^ 
chanics ;  viz.,  the  Cord,  Lever,  Inclined  Plane,  Pulley,  Screw,  Wheel  and 
Axle,  and  Wedge.  The  Cord,  Lever,  and  Inclined  Plane  are  called 
Simple  Machines  ;  the  others,  being  combinations  of  these,  are  called 
Compound  Machines. 

§  303. — In  Machines,  as  in  all  other  bodies,  every  action  is  ac- 
companied by  an  equal  and  contrary  reaction.  A  force  which  acts 
upon  a  Machine  to  impress  or  preserve  motion  is  called  a  Power. 
A  force  which  reacts  to  prevent  or  destroy  motion,  is  called  a 
Resistance.  The  Agent  which  is  the  source  of  power,  is,  §  38,  called 
a  Motor. 

§  304. — Resuming  Equation  (30),  and  supposing  the  displacement, 
which  in  that  equation  was  wholly  arbitrary,  to  conform  in  every 
respect  to  t^at  caused  by  the  powers  and  resistances,  we  shall  have 
5  s  =  d s,  s  being  the  path  described  by  the  elementary  mass  m; 
and  hence, 

2  Po  »  —  2  Ml  •  — —  .  rfs  =  0  ; 
a  r 


but 


whence, 


d^s  ds     d^s 

-— -  d  S  =z   —-  •  — -—    =   V  dv   z=  i  rf  1'2  ; 

di^  dt      dt  i         ■ 


2  P  5;)  —  ^  2  m  .  rf  r2  =  0. (506) 


346  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

Denoting  by  Q,  Q\  &c.  the  resistances,  by  P,  F',  &c.  the  pow- 
ei's,  S  q,  &c.  and  ^^j»,  &c.  the  projections  of  their  respective  virtual 
velocities  ;  the  first  term,  which  embraces  all  the  forces  except 
inertia  in  action  on  the  machine,  may  be  replaced  by  SPSj)  —  I.  QSq^ 
and  we   have 

j:P§2^  —  iQoq  =  ^^jn.dv^.    •     •     •     •     (507) 
Integrating, 

f^P8p-f^QSq  =  ^lmv^+  C; 

and   denoting   by  v^  the   initial   velocity,    and    taking    the    integral    so 
as   to    vanish   when    t  =  0, 

fl,  P  §2^  —  fi:  QSq  =  1--E  m  v^  -  -^  2  m  v^^.    •  •  •  (508) 

The  products  P  Sp  and  Q  S  q  are  the  elementary  quantities  of 
work  performed  by  a  power  and  a  resistance  respectively,  in 
the  element  of  time  d  t ;  the  product  ^7ndv^  is  the  elementary 
quantity  of  work  performed  by  the  inertia,  or  one  half  the  incre- 
ment of  living  force  of  the  mass  m  in  this  time.  And  Equation 
(508)  shows  that  in  any  machine,  in  motion,  the  increment  of  the 
half  sum  of  the  living  forces  of  all  its  parts  is  always  equal  to 
the  excess  of  the  work  of  the  powers  or  motors  over  that  of  the 
resistances. 

g305. — If  the   machine    start   from   rest.  Equation  (508)  becomes 

flPSp  -  f^QSq  =  ^^77iv%  •     •     •     .     (509) 

and  as  the  second  member  is  essentially  positive,  the  work  of  the 
motors  must  exceed  that  of  the  resistances  embraced  in  the  term 
fl.QSq;  in  other  words,  the  inertia  will  oppose  the  motor  and 
act  as  a  resistance.  When  the  motion  becomes  uniform,  the  second 
member  will  be  constant;  from  that  instant  inertia  will  cease  to 
act,  and  the  subsequent  work  of  the  motor  will  be  equal  to  that 
of  the  resistances  as  long  as  this  motion  continues.  If  the  motion 
be  now  retarded,  the  second  member  will  decrease,  the  inertia  will 
act  with    the    power,  and    this    will    continue    till    the    machine    comes 


APPLICATIONS.  347 

to  rest,  and  the  excess  of  uork  of  the  Besis/ance  during  retardatinu 
will  be  exactly  equal  to  that  of  the  J-'oiccr  during  acceleration. 
Generally,  then,  when  a  machine  is  at  rest  or  is  moving  uniformly, 
inertia  does  not  act ;  -when  the  motion  is  variable,  it  does,  and 
opposes  or  aids  the  motor  according  as  the  motion  is  accelerated 
or  retarded. 

g30G. — The  essential  parts  of  every  machine  are  those  which 
receive  directly  the  action  of  the  motor,  those  which  act  directly 
upon  the  body  to  be  moved  or  transformed,  and  those  which  serve 
to  transmit  the  action.  The  arrangement  of  the  latter  is  often  a 
source  of  resistance,  arising  from  Friction,  Adhesion,  Stiffness  of 
Cordage,    &c.,    whose    work    enters    largely    into    the    general    term 


/ 


FRICTION. 


g307. — When  two  bodies  are  pressed  together,  experience  shows 
that  a  certain  effort  is  always  required  to  cause  one  to  roll  or  slide 
along  the  other.  This  arises  almost  entirely  from  the  inequalities  in 
the  surfaces  of  contact  interlocking  with  each  other,  thus  rendering 
it  necessary,  when  motion  takes  place,  either  to  break  them  off,  com- 
press them,  or  force  the  bodies  to  separate  far  enough  to  allow  them 
to  pass  each  other.  This  cause  of  resistance  to  motion  is  called  fric- 
tion, of  which  we  distinguish  two  kinds,  according  as  it  accompanies 
a  sliding  or  rolling  motion.  The  first  is  denominated  sliding,  and 
the  second  rolling  friction.  They  are  governed  by  the  same  laws ; 
the  former  is  much  greater  in  amount  than  the  latter  under  given 
circumstances,  and  being  of  more  importance  in  machines,  will  prin- 
cipally occupy  our  attention. 

The  intensity  of  friction,  in  any  given  case,  is  measured  by  the 
force  exerted  in  the  direction  of  the  surface  of  contact,  which  will 
place  the  bodies  in  a  condition  to  resist,  during  a  change  of  state, 
in  respect  to  motion  or  rest,  only  by  their  inertia. 

1 308. — The  friction  between  two  bodies  maybe  measured  directly 
by  means  of  the  spring  balance.       For  this   purpose,  let  the  surface 


'yw'y. 


my. 


348  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

(7Z>  of  one   of  the    bodies  M  be   made   perfectly  level,  so    that    the 

other  body  M\  when  laid 

upon   it,   may  press    with 

its  entire  weight.   To  some 

point,  as  E^  of   the   body 

M',  attach  a  cord  with  a 

spring     balance     in      the 

manner  indicated  in  the  figure,  and    apply  to  the  latter  a  force  F  of 

such  intensity  as  to  produce  in  the  body  M  a  uniform  motion.      The 

motion    being    uniform,  the  accelerating  and  retarding  forces  must  be 

equal    and    contrary;    that  is  to  say,  the  friction    must  be  equal  and 

contrary  to   the  force  F^  of  which    the    intensity  is   indicated   by  the 

balance. 

The  experiments  on  friction  which  seem  most  entitled  to  confi- 
dence are  those  performed  at  Metz  by  M.  Morin,  under  the  orders 
of  the  French  government,  in  the  years  1831,  1832,  and  1833.  They 
were  made  by  the  aid  of  a  contrivance,  first  suggested  by  M.  Pon- 
celet,  which  is  one  of  the  most  beautiful  and  valuable  contributions 
that  theory  has  ever  made  to  practical  mechanics.  Its  details  are 
given  in  a  work  by  M.  Morin,  entitled  '•'•  Nouvelles  Experiences  sur  le 
Frottement:'     Paris,  1833. 

The  following  conclusions  have  been  drawn  from  these  experi- 
ments, viz. : 

The  friction  of  two  surfaces  which  have  been  for  a  considerable 
time  in  contact  and  at  rest  is  not  only  different  in  amount,  but  also 
in  nature,  from  the  friction  of  surfaces  in  continuous  motion;  espe- 
cially in  this,  that  the  friction  of  quiescence  is  subjected  to  causes  of 
variation  and  uncertainty  from  which  the  friction  during  motion  is 
exempt.  This  variation  does  not  appear  to  depend  upon  the  extent 
of  the  surface  of  contact ;  for,  with  different  pressures,  the  ratio  of 
the  friction  to  the  pressure  varied  greatly,  although  the  surfaces  of 
contact  were  the  same. 

The  slightest  jar  or  shock,  producing  the  most  imperceptible 
movement  of  the  surfaces  of  contact,  causes  the  friction  of  quies- 
cence to  pass  to  that  which  accompanies  motion.  As  every  machme 
may  be  regarded  as  being  subject  to  slight  shocks,  producing  imper- 


APPLICATIONS.  349 

ceptible  motions  in  the  surfaces  of  contact,  the  kind  of  friction  to  be 
employed  in  all  questions  of  equilibrium,  as  well  as  of  motions  of 
machines,  should  obviously  be  this  last  mentioned,  or  that  which 
accompanies  continuous  motion. 

The  LAWS  of  friction  which  accompanies  continuous  motion  are 
remarkably  uniform  and  definite.     These  laws  are : 

1st.  Friction  accompanying  continuous  motion  of  two  surfaces, 
between  which  no  unguent  is  interposed,  bears  a'  constant  proportion 
to  the  force  by  which  those  surfaces  are  pressed  together,  whatever 
be  the  intensity  of  the  force. 

2d.  Friction  is  wholly  independent  of  the  extent  of  the  surfaces  in 
contact. 

3d.  Where  unguents  are  interposed,  a  distinction  is  to  be  made 
between  the  case  in  which  the  surfaces  are  simply  unctudus  and  in 
intimate  contact  with  each  other,  and  that  in  Avhich  the  surfaces  are 
wholly  separated  from  one  another  by  an  interjjosed  stratum  of  the 
unguent.  The  friction  in  these  two  cases  is  not  the  same  in  amount 
under  the  same  pressure,  although  the  law  of  the  independence  of 
extent  of  surflice  obtains  in  each.  When  the  pressure  is  increased 
sufficiently  to  press  out  the  unguent  so  as  to  bring  the  unctuous  sur- 
faces in  contact,  the  latter  of  these  cases  passes  into  the  first;  and 
this  fact  may  give  rise  to  an  apparent  exception  to  the  law  of  the 
independence  of  the  extent  of  surftice,  since  a  diminution  of  the  sur- 
face of  contact  may  so  concentrate  a  given  pressure  as  to  remove  the 
unguent  from  between  the  surfaces.  The  exception  is,  however,  but 
apparent,  and  occurs  at  the  passage  from  one  of  the  cases  above- 
named  to  the  other.  To  this  extent,  the  law  of  independence  of  the 
extent  of  surfiice  is,  therefore,  to  be  received  with  restriction. 

There  are,  then,  three  conditions  in  respect  to  friction,  under 
which  the  surfaces  of  bodies  in  contact  may  be  considered  to  exist, 
viz.:  1st,  that  in  which  no  unguent  is  present;  2d,  that  in  which 
the  surfaces  are  simply  unctuous;  3d,  that  in  which  there  is  an 
interposed  stratum  of  the  unguent.  Throughout  each  of  these  states 
the  friction  which  accompanies  motion  is  always  proportional  to  the 
pressure,  but  for  the  same  pressure  in  each,  very  different  in 
amount. 


350 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


4th.  The  friction  which  accompanies  motion  is  always  independ- 
ent of  the  velocity  with  which  the  bodies  move;  and  this,  whether 
the  surfaces  be  without  unguents  or  lubricated  with  water,  oils, 
grease,  glutinous  liquids,   syrups,  pitch,   &c.,   &c. 

The  variety  of  the  circumstances  under  which  these  laws  obtain, 
and  the  accuracy  with  which  the  phenomena  of  motion  accord  with 
them,  may  be  inferred  from  a  single  example  taken  from  the  first 
set  of  Morin's  experiments  upon  the  friction  of  surfaces  of  oak, 
whose  fibres  were  parallel  to  the  direction  of  the  motion.  The  sur- 
faces of  contact  were  made  to  vary  in  extent  from  1  to  84;  the 
forces  which  pressed  them  together  from  88  to  2205  pounds ;  and 
the  velocities  from  the  slowest  perceptible  motion  to  9,8  feet  a 
second,  causing  them  to  be  at  one  time  accelerated,  at  another 
uniform,  and  at  another  retarded ;  yet,  throughout  all  this  wide 
range  of  variation,  in  no  instance  did  the  ratio  of  the  pressure  to 
the  friction  differ  from  its  mean  value  of  0,478  by  more  than  gV 
of  this    same   fraction. 

Denote  the  constant  ratio  of  the  normal  pressure 'P,  to  the  en- 
tire friction  F^  by  /;  then  will  the  first  law  of  friction  be  expressed 
by  the  following  equation, 


F 


(510) 


whence, 


F^f.P. 

This  constant  ratio  /  is  called  the  co-efficient  of  friction^  because, 
when  multiplied  by  the  total  normal  pressure,  the  product  gives 
the    entire  friction. 

Assuming  the  first  law  of  fric- 
tion, the  co-efficient  of  friction  may 
easily  be  obtained  by  means  of  the 
inclined  plane.  Let  W  denote  the 
weight  of  any  body  placed  upon 
the-  inclined  plane  A  B.  Resolve 
this  weight  G  G'  into  two  compo- 
nents, one  G  M  perpendicular  to 
the   plane,  and   the  other   G  N  par- 


APPLICATIONS.  351 

allel   to   it.      Because  the   angles    G' G M  and   BAC  are   equal,  the 
first   of  these   components  \vill  be 

G2I  =  W.cosA, 
and   the   second, 

GN  =  W.smA, 

in  which  A    denotes    the    angle  BA  C. 

The  first  of  these  components  determines  the  total  pressure  upon 
the   plane,  and    the   friction  due    to    this   pressure  will   be 

F  =f.  W  cos  A. 

The  second  component  urges  the  body  to  move  down  the  plane. 
If  the  inclination  of  the  plane  be  gradually  increased  till  the  body 
move  with  uniform  motion,  the  total  friction  and  this  component 
must  be   equal   and   opposed ;   hence, 

/.  W .  cos  A  =1  W .  sin  A  ; 

whence, 

.       sin  A  , 

f  = =  tan  A. 

''        cos^ 

We,  therefore,  conclude,  that  the  unit  or  co-efficient  of  friction 
between  any  two  surfaces,  is  equal  to  the  tangent  of  the  angle 
which  one  of  the  surfaces  must  make  with  the  horizon  in  order 
that  the  other  may  slide  over  it  Avith  a  uniform  motion,  the  body 
to  which  the  moving  surface  belongs  being  acted  upon  by  its  own 
weight  alone.  This  angle  is  called  the  angle  of  friction  or  limiting 
angle   of  resistance. 

The  values  of  the  nnit  of  friction  and  of  the  limiting  angles  for 
many  of  the  various  substances  employed  in  the  art  of  construction, 
are  given  in  Tables  VI,  VII,  and  VIII. 

Tlie  distinction  between  the  friction  of  surfaces  to  which  no  un- 
guent is  applied,  those  which  are  merely  unctuous,  and  those  between 
which  a  uniform  stratum  of  the  unguent  is  interposed,  appears  first 
to  have  been  remarked  by  M.  Morin  ;  it  has  suggested  to  him 
what  appears  to  be  the  true  explanation  of  the  difference  between 
his   results    and   those   of  Coulomb.      He   conceives,  that   in    the   ex- 


352  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

periments  of  this  celebrated  Engineer,  the  requisite  precautions  had 
not  been  taken  to  exclude  unguents  from  the  surfaces  of  contact. 
The  slightest  unctuositj,  such  as  might  present  itself  accidentally, 
unless  expressly  guarded  against — such,  for  instance,  as  might  have 
been  left  by  the  hands  of  the  workman  who  had  given  the  last 
polish  to  the  surfaces  of  contact — is  sufficient  materially  to  affect 
the   co-efficient  of  friction. 

Thus,  for  instance,  surfaces  of  oak  having  been  rubbed  with  hard 
dry  soap,  and  then  thoroughly  wiped,  so  as  to  show  no  traces 
whatever  of  the  unguent,  were  found  by  its  presence  to  have  lost 
|-'^«  of  their  friction,  the  co-efficient  having  passed  from  0,478 
to  0,164. 

This  effect  of  the  unguent  upon  the  friction  of  the  surfaces  may 
be  traced  to  the  fact,  that  their  motion  upon  one  another  without 
unguents  was  always  found  to  be  attended  by  a  wearing  of  both  the 
surfaces ;  small  particles  of  a  dark  color  continually  separated  from 
them,  which  it  was  found  from  time  to  time  necessary  to  remove, 
and  which  manifestly  influenced  the  friction  :  now,  vrith  the  presence 
of  an  unguent  the  formation  of  these  particles,  and  the  consequent 
wear  of  the  surfaces,  completely  ceased.  Instead  of  a  new  surface 
of  contact  being  continually  presented  by  the  wear,  the  same  surface 
remained,  receiving   by  the  motion  continually  a  more  perfect  polish. 

A  comparison  of  the  results  enumerated  in  Table  VIII,  leads  to 
the  following  remarkable  conclusion,  easily  fixing  itself  in  the  memory, 
that  with  the  unguents^  hogs'  lard  and  olive  oil  interposed  in  a  con- 
tinuous stratum  between  them,  surfaces  of  tvood  on  metal,  tvood  on 
wood,  metal  on  wood,  and  metal  on  metal,  when  in  motion,  have  all 
of  them  very  nearly  the  same  co-efficient  of  friction,  the  value  of  that 
co-efficient  being  in  all  cases  included  between  0,07  and  0,08,  and  the 
limiting   angle   of  resistance  therefore  between  4°  and  4°  35'. 

For  the  unguent  tallow  the  co-efficient  is  the  same  as  the  above  in 
every  case,  except  in  that  of  vietals  Ujjon  metals;  this  unguent  seems 
less  suited  to  metallic  surfaces  than  the  others,  and  gives  for  the 
mean  value  of  its  co-efficient  0,10,  and  for  its  limiting  angle  of  re- 
sistance 5°  43'. 


APPLICATIONS. 


553 


309. — Besides  friction,  there  is  another  cause  of  resistance  to  the 
motion  of  bodies  when  moving  over  one  another.  The  same  forces 
which  hold  the  elements  of  bodies  together,  also  tend  to  keep  the 
bodies  themselves  together,  when  brought  into  sensible  contact.  The 
effort  by  which  two  bodies  are  thus  united,  is  called  the  force  of 
Adhesion. 

Familiar  illustrations  of  the  existence  of  this  force  are  furnished 
by  the  pertinacity  with  which  sealing-wax,  wafei-s,  ink,  chalk,  and 
black-lead  cleave  to  paper,  dust  to  articles  of  dress,  paint  to  the 
surface    of  wood,  whitewash    to    the    walls    of  buildings,  and   the  like. 

The .  intensity  of  this  force,  arising  as  it  does  from  the  affinity 
of  the  elements  of  matter  for  each  other,  must  vary  with  the  num- 
ber of  attracting  elements,  and  therefore  with  the  extent  of  the  sur- 
face of  contact. 

This  law  is  best  verified,  and  the  actual  amount  of  adhesion  be- 
tween different  substances  determined,  by  means 
of  a  delicate  spring-balance.  Por  this  purpose, 
the  surfaces  of  solids  are  reduced  to  polished 
planes,  and  pressed  together  to  exclude  the  air, 
and  the  efforts  necessary  to  separate  them  noted 
by  means  of  this  instrument.  The  experiment 
being  often  repeated  with  the  same  substances, 
having  different  extent  of  surfaces  in  contact,  it 
is  found  that  the  effort  necessary  to  produce 
the  separation  divided  by  the  area  of  the  surface 
gives  a  constant  ratio.  Thus,  let  S  denote  the 
area  of  the  surfaces  of  contact  expressed  in  sfpiare 
feet,  square  inches,  or  any  other  superficial  unit; 
A  the  effort  required  to  separate  them,  and  a 
the   constant   ratio  in  question,  then  will 

A 


or. 


A  -  a.  S. 


The    constant   a   is   called    the   tinit   or  co-efficient  of  adhesion,  and  ob- 


354 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


viouslj  expresses  the  value  of  adhesion  ou  each  unit  of  sui'face,  for 
making 

we   have 

A  z=  a. 

To  find  the  adhesion  between  solids  and  liquids,  suspend  the  solid 
from  the  balance,  with  its  polished  surface  doAvnward  and  in  a  hori- 
zontal position  ;  note  the  weight  of  the  solid, 
then  bring  it  in  contact  with  the  horizontal 
surface  of  the  fluid  and  note  the  indication  of 
the  balance  when  the  separation  takes  place, 
on  drawing  the  balance  up  ;  the  difference  be- 
tween this  indication  and  that  of  the  weight 
will  give  the  adhesion ;  and  this  divided  by 
the  extent  of  surface,  will  give,  as  before,  the 
co-efficient  a.  But  in  this  experiment  two 
opposite  conditions  must  be  carefully  noted, 
else  the  cohesion  of  the  elements  of  the  liquid 
for  each  other  may  be  mistaken  for  the  adhe- 
sion of  the  solid  for  the  fluid.  If  the  solid 
on  being  removed  take  with  it  a  layer  of  the 
fluid ;    in    other    words,    if    the    solid    has   been 

wet  by  the  fluid,  then  the  attraction  of  the  elements  of  the  solid 
for  those  of  the  liquid  is  stronger  than  that  of  the  elements  of  the 
liquid  for  each  other,  and  a  will  be  the  unit  of  adhesion  of  two 
surfaces  of  the  fluid.  If,  on  the  contrary,  the  solid  on  leaving  the 
fluid  be  perfectly  dry,  the  elements  of  the  fluid  will  attract  each 
other  more  powerfully  than  they  will  those  of  the  solid,  and  a  will 
denote   the   unit   of  adhesion  of  the    solid    for    the   liquid. 

It  is  easy  to  multiply  instances  of  this  diversity  in  the  action  of 
solids  and  fluids  upon  each  other.  A  drop  of  water  or  spirits  of 
wine,  placed  upon  a  wooden  table  or  piece  of  glass,  loses  its  globu- 
lar form  and  spreads  itself  over  the  surface  of  the  solid ;  a  drop  of 
mercury  will  not  do  so.  Immerse  the  finger  in  water,  it  becomes 
wet ;    in    quicksilver,  it   remains    dry.     A   tallow  candle,  or  a  feather 


APPLICATIONS. 


355 


from  any  species  of  water-fuwl,  remains  dry  though  dipped  in  water. 
Gold,  silver,  tin,  lead,  &c.,  become  moist  on  being  immersed  in 
quicksilver,  but  iron  and  platinum  do  not.  Quicksilver  when  poured 
into  a  gauze  bag  will  not  run  through  ;  water  will :  place  the  gauze 
containing  the  quicksilver  in  contact  with  water,  and  the  metal  will 
also  flow  through. 

It  is  difficult  to  ascertain  the  precise  value  of  the  force  of  adhe- 
sion between  the  rubbing  surfaces  of  machinery,  apart  from  that  of 
friction.  But  this  is  attended  with  little  practical  inconvenience,  as 
long  as  a  machine  is  in  motion.  The  experiments  of  which  the 
results  are  given  in  Tables  VI,  VII,  and  VIII,  and  which  are  applicable 
to  machinery,  were  made  under  considerable  pressures,  such  as  those 
with  which  the  parts  of  the  larger  machines  are  accustomed  to  move 
upon  one  another.  Under  such  pressures,  the  adhesion  of  unguents 
to  the  surfaces  of  contact,  and  the  opposition  to  motion  presented 
by  their  viscosity,  are  causes  whose  influence  may  be  safely  disre 
garded  as  compared  with  that  of  friction.  In  the  cases  of  lighter 
niachinery,  however,  such  as  watches,  clocks,  and  the  like,  these 
considerations  rise    into    importance,  and    cannot   be    neglected. 


STIFFNESS    OF    CORDAGE. 

§  310. — Conceive  a  wheel  turning 
freely  about  an  axle  or  trunnion,  and 
having  in  its  circumference  a  groove  to 
receive  a  cord  or  rope.  A  weight  IF, 
being  suspended  from  one  end  of  the 
rope,  while  a  force  h\  is  applied  to  the 
other  extremity  to  draw  it  up,  the 
latter  will  experience  a  resistance  in 
consequence  of  the  rigidity  of  the  rope, 
which  opposes  every  effort  to  bend  it 
around  the  wheel.  This  resistance  must, 
of  necessity,  consume  a  portion  of  the 
wo;^  of  the  force  F.  The  measure  of 
the   resistance    due    to    the    rigidity  of    cordage    has    been    n 


ladc    the 


356  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

subject  of  experiment  by  Coulomb ;  and,  according  to  him,  it 
results  that  for  the  same  cord  and  same  wheel,  this  measure  is 
composed  of  two  parts,  of  which  one  remains  constant,  while  the 
other  varies  with  the  weight  W,  and  is  directly  proportional  to  it; 
so  that,  designating  the  constant  part  by  IT,  and  the  ratio  of  the 
variable  part  to  the  weight  W  by  /,  the  measure  will  be  given  by 
the  expression 

A'+  /.  W; 

in  which  K  represents  the  stiffness  arising  from  the  natural  torsion 
or  tension  of  the  threads,  and  /  the  stiffness  of  the  same  cord  due  to 
a  tension  resulfmg  from  one  unit  of  weight ;  for,  making  W  =  1,  the 
above  becomes 

K+  I. 

Coulomb  also  found  that  on  changing  the  wheel,  the  stilTness  varied 
in  the  inverse  ratio  of  its  diameter ;   so  that  if 

K+  I.  W 

be  the  measure  of  the  stiffness  for  a  wheel  of  one  foot  diameter,  then 

will 

K-{-  I.W 
2E 

be  the  measure  when  the  wheel  has  a  diameter  of  2  R.  A  table 
giving  the  values  of  K  and  /  for  all  ropes  and  cords  employed  in 
practice,  when  wound  around  a  wheel  of  one  foot  diameter,  and  sub- 
jected to  a  tension  arising  from  a  unit  of  weight,  would,  therefore, 
enable  us  to  find  the  stiffness  answering  to  any  other  wheel  and 
weight  whatever. 

But  as  it  would  be  impossible  to  anticipate  all  the  different  sizes 
of  ropes  used  under  the  various  circumstances  of  practice,  Coulomb 
also  ascertained  the  law  which  connects  the  stiffness  with  the  diame- 
ter of  the  cross-section  of  the  rope.  To  express  this  law  in  all  cases, 
he  found  it  necessary  to  distinguish,  1st,  new  white  rope,  either  dry 
or  moist ;  2d,  lohite  ropes  partly  worn,  either  dry  or  moist ;  3d,  tarred 
ropes  ;  4th,  packthread.  The  stiffness  of  the  first  class  he  found  ngarly 
proportional  to  the  square  of  the  diameter  of  the  cross-section  ;    that 


APPLICATIONS.  357 

of  the  second,  to  the  square  root  of  the  cube  of  this  diameter,  nearly ; 
that  of  the  third,  to  the  number  of  yarns  in  the  rope;  and  that  of 
the  fourth,  to  the  diameter  of  the  cross-section.  So  that,  if  S  denote 
the  resistance  due  to  the  stiffness  of  any  given  rope ;  d  the  ratio  of 
its  diameter  to  that  of  the  table ;  and  n  the  ratio  of  the  number  of 
yarns  in  any  tarred  rope  to  that  of  the  table,   we  shall  have  for 

^Vcifl  white  rope,  dry  or  moist. 

^^'^  ■  —2ir- (^") 

Half  leorn  ichitc  rope,  dry  or  moist, 

^        ,1    K  +  I.W  ,       ^ 

^  =  '" — ¥jr- (512) 

Tarred  rope. 
Packthread. 

For  packthread,  it  will  always  be  sufficient  to  use  the  tabular 
values  given,  corresponding  to  the  least  tabular  diameters,  and  substi- 
tute them  in  Equation  (514).  An  example  or  two  will  be  sufficient 
to  illustrate  the  use  of  these  tables. 

E.vample  \st.  Required  the  resistance  due  to  the  stiffness  of  a  new 
dry  white  rope,  whose  diameter  is  1,18  inches,  when  loaded  with 
a  weight  of  882  pounds,  and  wound  about  a  wheel  1,64  feet  in 
diameter. 

Seek  in  No.  1,  Table  IX,  the  diameter  nearest  that  of  the  given 
rope  ;    it  is  0,79  ;    hence, 

and  from  the  table  at  the  side. 

From  No.   1,  opposite  0,79,  we  find 

K  -  1.0097, 
/   =  0,03195; 


358    ELEMENTS  OF  ANALYTICAL  MECHANICS. 

ft- 
which,  together    with    the    weight    W  =  883   lbs.,   and   2  B  =  1,64, 

substituted  in  Equation  (511),  give 

lb.  lb. 

S  =  3,25  .  '.609^  +  0.03'«'  X  882  ^  ^ -,^_ 

which    is    the    true    resistance   due   to   the   stiffness   of   the   rope   in 
question. 

Example  2d.  What  is  the  resistance  due  to  the  stiffness  of  a 
white  rope,  half  worn  and  moistened  with  water,  having  a  diam- 
eter equal  to  1,97  inches,  wound  about  a  wheel  0,82  of  a  foot  in 
diameter,  and  loaded  with  a  weight  of  2205  pounds'? 

The  tabular  diameter  in  No.  4,  Table  IX,  next  less  than  1,97, 
is  1,57,  and  hence, 

d  =  Y^  =  l,d  nearly; 

the  square  root  of  the  cube  of  which  is,  by  the  table  at  the  side, 

d^  =  1,482. 
In  No.  4  we  find,  opposite   1,57, 

K  =  6,4324, 

/    =  0,06387  ; 

ft. 
which    values,    together    with    W  =  2205   lbs.,   and   2  H  =  0,82,   in 

Equation  (512),  give 

lbs .  lbs. 

S  =  :,483  X  M324  +  0,06387  X  2205  ^  ^^ -^^^ 

which  is  the  required  resistance. 

Example  Sd.  What  is  the  resistance  due  to  the  stiffness  of  a 
tarred  rope  of  22  yarns,  when  subjected  to  the  action  of  a  weight 
equal  to  4212  pounds,  and  wound  about  a  wheel  1,3  feet  diameter, 
the  weight  of  one  running  foot  of  the  rope  being  about  0,6  of  a 
pound  1 

By  referring  to  No.  5,  Table  IX,  we  find  the  tabular  number  of 
yarns  next  less  than  22  to  be  15,  and  hence, 

22 
n  =  "^  =  1,466  nearly. 
15 


APPLICATIONS. 


359 


In  the  same  table,  opposite  15,  we  find 

K  =  0,7G64, 

/    =  0,019879; 

ft- 
which,  together  with   W=  4212,  and  2  Ji  =  1,3,  in  Equation  (513), 

give 

0,7664  +  0,019879  x  4212 


S  =  1,466 


1,3 


lbs. 

=  95,188. 


Example  4:th.  Required  the  resistance  due  to  the  stiffness  of  a 
new  white  packthread,  whose  diameter  is  0,190  inches,  when  moist- 
ened or  wet  with  water,  woimd  about  a  wheel  0,5  of  a  foot  in 
diameter,   and  loaded  with  a  weight  of  275  pounds. 

The   lowest   tabular   diameter   is  0,39  of  an   inch,  and   hence 

,       0.196 

^  =  0;390  =^^^^'"'^^'- 

In   No.  2,  Table  IX,  we  find,  opposite  0,39, 

K  -  0,8048, 
/  =  0,00798 ; 

which,  with   W  =  275,  and   2Ii  —  0,5,  we  find,  after   substituting  in 
Equation  (514), 


0,5 


0,8048  +  0,00798  x  275 

"    ~    o;5 


=  2.999. 


§311. — The  resistance  just  found 
is  expressed  in  pounds,  and  is  the 
amount  of  weight  which  would  be 
necessary  to  bend  any  given  rope 
around  a  vertical  wheel,  so  that 
the  portion  A  £,  between  the  first 
point  of  contact  A,  and  the  point 
U,  where  the  rope  is  attached  to 
the  weight,  shall  be  perfectly  straight. 
The  entire  process  of  bending  takes 
place  at  this  first  or  tangential 
point    A  ;    for,  if  motion    be   com- 


360         ELEMENTS     OF    ANALYTICAL    MECHANICS. 

municated  to  the  wheel  in  the  direction  indicated  by  the  arro-w- 
head,  the  rope,  supposed  not  to  slid^  will,  at  this  point,  take  and 
retain  the  constant  curvature  of  the  wheel,  till  it  passes  from  the 
latter  on  the  side  of  the  power  F,  When,  therefore,  by  the  motion 
of  the  wheel,  the  point  m  of  the  rope,  now  at  the  tangential  point, 
passes  to  to',  the  working  point  of  the  force  S  will  have  described 
in  its  own  direction  the  distance  A  D.  Denoting  the  arc  described 
by  a  point  at  the  unit's  distance  from  the  centre  of  the  wheel 
by  s, ,  and  the   radius  of  the   wheel   by  i2,  we   shall   have 

ADz=Rs,', 

and  representing   the   quantity  of  work  of  the  force  S  by  Z,  we  get 

replacing  S  by  its  value  in  Equations  (511)  to  (514), 

L  =  Rsrdr^^'    .....    (515) 

3 

in  which  d^  represents  the  quantity  d-,  d^,  n,  or  c?,  in  Equations  (511) 
to  (514),  according    to   the   nature  of  the    rope. 

Example.— Tskmg  the  2d  example  of  §310,  and  supposing  a  por- 
tion of  the  rope,  equal  to  20  feet  in  length,  to  have  been  brought 
in   contact  with   the   wheel,  after   the   motion   begins,  we  shall  have 

X  =  20  X  206,109  =  5322,18   units  of  work; 

that  is,  the  quantity  of  work  consumed  by  the  resistance  due  to 
the  stiffness  of  the  rope,  while  the  latter  is  moving  over  a  distance 
of  20  feet,  would  be  sufficient  to  raise  a  weight  of  5322,18  pounds 
through   a  vertical   height  of  one  foot. 

FRICTION   ON   PIVOTS,   AND  TRUNNIONS. 

§312. — All  rotating  pieces,  such  as  wheels  supported  upon  other 
pieces,  give  rise  by  their  motion  to  friction.  This  is  an  important 
element  in  all  computations  relating  to  the  performance  of  machinery. 
It   seems    to    be   different   according   as   the    rotating  .pieces  are    kept 


APPLICATIONS. 


5G1 


ill  place  by  trunnions  or  by 
pivots.  By  trunnions  are  meant 
cylindrical  projections  a  a  from 
the  ends  of  the  arbor  yl ^  of  a 
wheel.  The  trunnions  rest  on  the 
concave  surfaces  of  cylindrical 
boxes  CD,  with  which  they  usu- 
ally have  a  small  surface  of 
contact  "TO,  the  linear  elements 
of  both  being  parallel.  Pivots 
are  shaped  like  the  trunnions, 
but  support  the  weight  of  the 
wheel  and  its  arbor  upon  their 
circular  end,  which  rests  against 
the  bottom  of  cylindrical  sock- 
ets FGIII. 


7  ,---v     J? 


tisSl 


pwi 


.:-;^^s^^ig^:mM^^ 


Let  iV"  denote  the  force,  in  the  direction  of  the  axis,  by  which 
the  pivot  is  pressed  against  the 
bottom  of  the  socket.  This  force 
may  be  regarded  as  passing 
through  the  centre  of  the  cir- 
cular end  of  the  pivot,  and  as 
the  resultant  of  the  partial  pres- 
sures exerted  upon  all  the  ele- 
mentary surfaces  of  which  this 
circle  is  composed.  Denote  by 
A  the  area  of  the  entire  circle, 
then  will  the  pressure  sustained 
by  each  unit  of  surface  be 

A' 

and    the   pressure  on  any  small  portion  of   the  surface  denoted  by  a, 
will  obviously  be 


362 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


and  the  friction  on  the  same  will  be 


This  friction  may  be  regarded  as  applied  to  the  centre  of  the  ele- 
mentary surface  a;  it  is  opposed  to  the  motion,  and  the  direction  of 
its  action  is  tangent  to  the  circle  described  by  the  centre  of  the 
element.  Denote  the  radius  of  this  circle  by  .r,  then  will  the  mo- 
ment of   the  friction  be 

Now,  if  .«  denote  the  length  of  any  variable  portion  of  the  circumfer- 
ence  at  the  unit's  distance  from  the  centre   (7,  then  will 


also. 


a  =z  X  .  d  s  .  d  X 


which  substituted  above  give 
and  by  integration, 


x"^  .  d  X  .  d 


f'N 


/R  /»2  f 

x"^  d  X   I       d  s 


nt  E^ 


=  f-N-iR', 


(510) 


whence  we  conclude,  that,  in  the  fric- 
tion of  a  pivot,  tve  may  regard  the 
xvUole  friction  due  to  the  pressure  as 
acting  in  a  single  point,  and  at  a  dis- 
tance from  the  centre  of  motion  equal 
to  two-thirds  of  the  radius  of  the  base 
of  the  pivot.  This  distance  is  called 
the  mean  lever  of  friction. 

§313. — If  the  extremity  of  the  pivot, 
instead  of  rubbing  upon  an  entire  circle, 
is  only  in  contact  with  a  ring  or  sur- 
face comprised    between  two  concentric 


APPLICATIONS.  363 

circles,  as   -when    the    arLor   of  a  wheel    is  urged   in    the  direction  of 
its  length  by  the  force  N  against  a  shoulder  deb  a;    then  will 

^  =  *  (722  -  B'-)  • 

and  the  integration  will  give 

in    which   R   denotes    the    radius    of    the    larger,  and  R'  that    of  the 
smaller  circle. 

Finally,    denote   by   I  the   breadth   of    the   ring,   that   is,    the   dis- 
tance A'  A;     by  r,  its    mean    radius  or  distance  from    C    to  a   point 


517) 


and  making 


we  obtain,  for  the  moment  of  the  friction  on  the  entire  ring, 

/-V.r, (518) 

The  quantity  r^  is  called  the  mean  lever  of  friction  for  a  ring.  Since 
the  whole  friction  fN  may  be  considered  as  applied  at  a  point 
whose  distance  trom  the  centre  is  f  jR,  or  r^  =  r  -\-  — — >  according 
as  the  friction  is  exerted  over  an  entire  circle  or  over  a  ring, 
and  since  the  path  described  by  this  point  lies  always  in  the  di- 
rection in  which  the  friction  acts,  the  quantity  of  work  consumed 
by  it  will  be  equal  to  the  product  of  its  intensity  fN  into  this 
path.  Designating  the  length  of  the  arc  described  at  the  upit's 
distance  from   C  by  s^ ,  the    path    in    question  will   be    either 

f  /?  s, ,     or     i\  s,  ; 


362         ELEMENTS     OF     ANALYTICAL    MECHANICS, 
and  the  friction  on  the  same  will  be 

A       ' 

This  friction  may  be  regarded  as  applied  to  the  centre  of  the  ele- 
mentary surface  a;  it  is  opposed  to  the  motion,  and  the  direction  of 
its  action  is  tangent  to  the  circle  described  by  the  centre  of  the 
element.  Denote  the  radius  of  this  circle  by  .r,  then  will  the  mo- 
ment of  the  friction  be 

Now,  if  .«  denote  the  length  of  any  variable  portion  of  the  circumfer- 
gj^g^  of  +iiQ  unit's  rlititnnpp  from  the  centre   C.  then  will 


alsc 


whi 


-  ^/c^[h  ^.i^J  =Mi^^A  -i^J ,. 


and  by  integration. 


f'N 


x^  d  X    I       d 


n<B? 


=  f'N'iR; 


(510) 


whence  we  conclude,  that,  in  the  fric- 
tion of  a  pivot,  we  may  regard  the 
whole  friction  due  to  the  ^:»rc5S!/?-e  as 
acting  in  a  single  j^oint,  and  at  a  dis- 
tance from  the  centre  of  motion  equal 
to  two- thirds  of  the  radius  of  the  base 
of  the  jiivot.  This  distance  is  called 
the  7nean  lever  of  friction. 

§313. — If  the  extremity  of  the  pivot, 
instead  of  rubbing  upon  an  entire  circle, 
is  only  in  contact  with  a  ring  or  sur- 
face comprised    between  two  concentric 


APPLICATIONS.  3G3 

circles,  as   Mhon    the    arLor    of  a  wheel    is  urged    in    the  direction  of 
its  length  by  the  force  X  against  a  shoulder  deb  a;    then  will 

A  =  '^  {R"-  -  i2'2)  ; 

and  the  integration  will  give 

in    which  Ji   denotes    the    radius    of    the    larger,  and  E'  that    of  the 
smaller  circle. 

Finally,  denote  by  I  the  breadth  of  the  ring,  that  is,  the  dis- 
tance A'  A ;  by  7;  its  mean  radius  or  distance  from  C  to  a  point 
half  way  between  A'  and  A,  and  we  shall  have 

R'  =  r  -  \l; 

substituting  these  values  above  and  reducing,  we  have 


/..Vx   [r  +  T^—]  ; (517) 


and  making 


12  r 


we  obtain,  for  the  moment  of  the  friction  on  the  entire  ring, 

/.i^^^ (51S) 

The  quantity  r^  is  called  the  mean  lever  of  friction  for  a  ring.  Since 
the  whole  friction  fN  may  be  considered  as  applied  at  a  point 
whose  distance  from  the  centre  is  -|  J?,  or  r^  =  r  -\-  —^^  according 
as  the  friction  is  exerted  over  an  entire  circle  or  over  a  ring, 
and  since  the  path  described  by  this  point  lies  always  in  the  di- 
rection in  which  the  friction  acts,  the  quantity  of  work  consumed 
by  it  will  be  equal  to  the  product  of  its  intensity  fN  into  this 
path.  Designating  the  length  of  the  arc  described  at  the  unit's 
distance  from   C  by  s^ ,  the   path    in    question  will   be    cither 

f  i?  s, ,     or     r^  s,  ; 


364  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and   the    quantity  of  work  either 


for   an   entire   circle,   or 

12  r> 


/•^('•  +  T^)'' 


for  a  ring.  Let  Q  denote  the  quantity  of  work  consumed  by  fric- 
tion in  the  unit  of  time,  and  n  the  number  of  revolutions  performed 
by  the  pivot  in   the   same   time  ;   then  will 

s^  r=  2  -n-  X  » ; 
and  we   shall    have 

Q  =  ^'TT.E.f.JV.n (519) 

for  the   circle,  and 

i!_ 
12r> 


Q  =  2fr-/'iy'(r  +  ^)  .n      ....     (520) 


I 

for    a   rins :   in  which  *  =  3.1416. 


'»  J 


The  co-efficient  of  friction  /,  when  employed  in  either  of  the  fore- 
going  cases,  must  be   taken  from  Table  VI,  VII,  or  VIII. 

Example. — Required  the  moment  of  the  friction  on  a  pivot  of 
cast  iron,  working  into  a  socket  of  brass,  and  which  supports  a 
weight  of  1784  pounds,  the  diameter  of  the  circular  end  of  the 
pivot  being  6  inches.     Here 

in.  ft. 

i2  =  f  =  3  =  0,25, 

lbs. 

iV  =  1784, 
/  =  0,147; 
which,  substituted    in  Equation  (516),  gives 

lbs.  ft. 

0,147  X  1784  X  f  X  0,25  =  43,708. 

And  to  obtain  the  quantity  of  work  in  one  unit  of  time,  say  a 
minute,  there  being  20  revolutions  in  this  unit,  we  make  n  =  20, 
and  If  =  3,1416  in  Equation  (519),  and  find 

Q  =  ^  X  3,1416  X  0,25  X  0,147  x  1784  x  20  =  5492,80 ; 


APPLICATIONS.  365 

that  is  to  say,  during  each  unit  of  time,  there  is  a  quantity  of 
work  lost  which  would  be  sufiicient  to  raise  a  weight  of  5492,80 
pounds  through   a  vertical   distance  of  one  foot. 

Example. — Required  the  moment  of  friction,  when  the  pivot  sup- 
ports a  weight  of  204G  pounds,  and  works  upon  a  shoulder  whose 
exterior  and  interior  diameters  arc  respectively  6  and  4  inches ;  the 
pivot   and   socket   being   of  cast   iron,  with  water   interposed, 

I  =  — ^ —  =  1  inch, 

r  =  2  -f  0,5  =  2,5  inches, 

(1)2  .n.  ft. 

r,  =  2,5  +  ,,^  \.  =  2,5333  =  0.2111, 
'  '  12  X  2,D  ' 

N  =  2046  pounds, 

/=  0,314; 

which,  substituted  in  E.xpression  (518),  gives  for  the  moment  of  friction, 

0,314  X  204g'x  0,2111  =  135,62. 

The  quantity  of  work  consumed  in  one  minute,  there  being  sup- 
posed 10  revolutions  in  that  unit,  will  be  found  by  making  in 
Equation  (520),  if  =  3,1416  and  n  =  10, 

Q  =  2  X  3,1416  x  0,314  x  2046  X  0,211  x  10  =  8517,24; 

that  is  to  say,  friction  Avill,  in  one  unit  of  time,  consume  a  quantity 
of  w^ork  which  would  raise  8517,24  pounds  through  a  vertical  dis- 
tance of  one  foot.  The  quantity  of  work  consumed  in  any  given 
time  would  result  from  multiplying  the  work  above  found,  by  the 
time   reduced   to   minutes. 

TRUNNIONS. 

1 314. — The  friction  on  trunnions  and  axles,  which  we  now  pro- 
ceed to  consider,  gives  a  considerably  less  co-efficient  than  that  which 
accompanies  the  kinds  of  motion  referred  to  in  §  308.  This  will 
appear  from  Table  X,  which  is   the   result  of  careful   experiment. 

The  contact  of   the    trunnion  with    its   box    is    alon^   a    linear  ele- 


366 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


meut,  common  to  the  surfaces  of  both.  A  section  perpendicular  to 
its  length  would  cut  from  the  trunnion  and  its  box,  two  circles  tan- 
gent to  each  other  internally.  The  trunnion  being  acted  on  only  by 
its  weight,  would,  when  at  rest,  give  this  tangential  point  at  o,  the 
lowest  point  of  the  section  f  o  q  o^  the  box.  If  the  trunnion  be  put 
in  motion  by  the  application  of  a  force,  it  would  turn  around  the 
point  of  contact  and  roll 
indefinitely  along  the  sur- 
face of  the  box,  if  the 
latter  were  level ;  but  this 
not  being  the  case,  it  will 
ascend  along  the  inclined 
surface  o  p  \.o  some  point 
as  ???,  where  the  inclina- 
tion of  the  tangent  u  m  v 
is  such,  that  the  friction 
is  just  sufficient  to  pre- 
vent the  trunnion  from  sliding.  Here  let  the  trunnion  be  in  equili- 
brio.  But  the  equilibrium  requires  that  the  resultant  of  all  the 
forces  which  act,  friction  included,  shall  pass  through  the  point  m 
and  be  normal  to  the  surface  of  the  trunnion  at  that  point.  The 
friction  is  ap^ilied  at  the  point  m ;  hence  the  resultant  N  of  all  the 
other  forces  must  pass  through  m  in  some  direction  as  md;  the 
friction  acts  in  the  direction  of  the  tangent;  and  hence,  in  order 
that  the  resultant  of  the  friction  and  the  force  N  shall  be  normal  to 
the  surface,  the  tangential  component  of  the  latter  must,  when  the 
other  component  is  normal,  be  equal  and  directly  oj^posed  to  the 
friction. 

Take  upon  the  direction  of  the  force  N  the  distance  in  d  to 
represent  its  intensity,  and  form  the  rectangle  adb  m,  of  which 
the  side  m  b  shall  coincide  with  the  tangent,  then,  denoting  the 
angle  dm  a  by  9,  will  the  component  of  N  perpendicular  to  the  tan- 
gent be 

N .  cos  9  ; 

and  the  friction  due  to  this  pressure  will  be 

f .  N .  cos  9. 


A  P  P  L  I  C  A  T  1 0  X  S  . 


367 


The  component  of  X,  in  the  direction  of  the  tangent,  will  be 

iV .  sin  (p  ; 

and  as  this  must  be  equal  to  the  friction,  we   have 

/.  iV.  cos  9  =  iV.  sin  (p  ; (5'1) 

whence, 

/  =  tan  (p  ; 

that  is  to  say,  the  ratio   of  the  friction    to    the  j^ressure  on    the    trun- 
nion   is  equal    to    the    tangent  of  the   angle    winch    the   direction    of  the 
resultant  JV,  of  all  the  forces   except   the  friction,  males   with    the   nor. 
mal    to    the   surface    of    the    trunnion    at 
the  jjoint  of  contact.     This  gives  an  easy 
method   of    finding    the    point    of   con- 
tact.      For   this   purpose,    we   have   but 
to    draw    through    the   centre    A   a   line 
A  Z,    parallel    to     the    direction    of    iV, 
and    through   A   the    line   A  m,  making 
with  A  Z   an   angle   of  which    the    tan- 
gent  is  /;    the  point   m,   in   which   this 
line    cuts     the    circular    section    of    the 

n 

trunnion,  will   be  the  point  of  contact. 

Because  m  a  d  i,   last  figure,  is  a  rectangle,  we  have 

iV2   =   iV"2  C0S2  (p   _^   JV2  siu2  ;p  ; 

and,  substituting  for  ^V2sin2(p   its  equal  /2  iV^  cos2  (p,   we  have 

JV2  =   iY2  C0S2  9  +  /2  iV2  C0S2  (p   =   jV^  C0s2  (p  (1  -f-  /2)  J 


whence, 


iVcOS  9   =:  iV  X 


VTT7 


and  multiplying  both  members  by  /, 

/.  ^V.  cos(?  =  i\^.  -- 


/ 


(5-2) 


but   the    first    member    is    the     total    friction  :    whence    we     conclude 
that   to  f  lid  the  friction   vjjon   a   trunnion,  we  have  but  to  multiply  the 


368 


ELEMENTS    OF    ANALYTICAL    MECHANICS. 


resultant  of  the  forces  which  act  upon  it  hy  the  unit  of  friction,  found 
in  Table  X,  and  divide  this  product  by  the  square  root  of  the  square 
of  this  same  unit  increased  by  unity. 

This  friction   acting  at  the  extremity  of  the  radius  R  of  the  trun- 
nion and  in  the  direction  of  the  tangent,   its  moment  will  be 


N' 


f 


/I  +f' 


X  R. 


(523) 


And  the  path  described  by  the   point  of  application  of   the   friction 
being  denoted  by  Rs^,  the  quantity  of  work  of  the  friction  will  be 


N .  R  .s.  X 


/ 


vTTy-' 


(524) 


in  which  s^  denotes  the  path  described  by  a  point  at  the  unit's  dis- 
tance from  the  centre  of  the  trunnion.  Denoting,  as  in  the  case  of 
the  pivot,  the  number  of  revolutions  performed  by  the  trunnion  in 
a  unit  of  time,  say  a  minute,  by  « ;  the  quantity  of  work  performed 
by  friction  in  this  time  by   Q, ;    and  making  -r  =  3,1416,  we  have 


s^  =  2  *  .  n  ; 


and 


Q^  z^'H'K  .R.n.N. 


f 


-/r+/2 


(525) 


When   the    trunnion   remains   fixed   and   docs    not    form   part   of    the 
rotating   body,   the   latter  will    turn    about   the   trunnion,    which    now 
becomes    an    axle,   having    the   centre   of 
motion  at  A,   the   centre    of   the   eye   of 
the  wheel ;  in  this  case,  the  lever  of  fric- 
tion  becomes   the   radius   of   the   eye   of 
the    wheel.      As    the    quantity   of   work 
consumed    by    friction     is     the     greater. 
Equation     (525),    in    proportion    as    this 
radius   is   greater,   and   as   the   radius   of 
the    eye    of    the  wheel    must    be    greater 

than  that  of  the  axle,  the  trunnion  has  the  advantage,  in  this  respect^ 
over  the  axle. 


APPLICATIONS. 


569 


The  value  of  the  quantity  of  work  consumed  by  friction  is  wholly 
independent  of  the  length  of  the  trunnion  or  axle,  and  no  advantage 
is  therefore  gained  by  making  it  shorter  or  longer. 


THE    CORD. 


§  315. — The  cord  and  its  properties  have  been  considered  in  part 
at  §  58.  It  is  now  proposed  to  discuss  its  action  under  the  opera- 
tion of  forces  applied  to  it  in  any  manner  whatever. 

Let  the  points  A',  A",  A'",  be  connected  with  each  other  by 
means  of  two  perfectly  flex- 
ible and  inextensible  cords 
A' A",  A"  A'",  the  first 
point  being  acted  upon  by 
the  forces  P',  P",  &c. ;  the 
second  by  the  forces  Q',  Q", 
&c. ;  and  the  third  by  the 
forces  S',  S",  &;c. ;  and  sup- 
pose these  forces  to  be  in 
equilibrio.  Denote  the  co- 
ordinates of  A'  by  x'y'z', 
A"hy  x"  !/"z",iindA"'hy 


hl•a,\c  sum  of  the  components  of  the  forces  acting  at  A'  in  the  direc- 
tion of  xyz,  by  X'  Y'  Z\  at  A"  by  X"  Y"  Z",  and  at  A'"  by 
X'"  Y'"  Z'".      Then  will,  §  101, 


X'    Sx'    -j-  Y'    St/    +  Z'    0  z' 
+  X"  S  x"    +  Y"  0  7j"   +  Z"  0  z"    \  =  0. 
-f  X"'dx"'  +  Y"'5t/"'  +  Z"'oz"' 

Denote  the  length  .I'.-l"  by/,  and  A"  A'"  by  ff ;    then  will 


L   =/-  v^"   _x')2+(y"    _y')2+(3"    -z'y::^0.   1 
R  =  ff-  ^{x"'  -  x'y  +  ii/'"  -  y"f  +  {z'"  -  z'y  =  0.    J 


(52G) 


(527) 


The    displacement    by  which  we    obtain    the    virtual    velocities    whose 

24 


370  ELEMENTS     OF    ANALYTICAL    MECHANICS. 


projections  are  S  x',  S  y\  S  z\  &c.,  is  not  wholly  arbitrary;  but  must 
be  made  so  as  to  satisfy  the  condition 

§/=  0    md    Sff  =  0. (528) 

Differentiating    Equations    (527),    and   writing    for    dx',    dy\    dz', 
S  x',  Oy',  Sz',  «Scc.,  we  find 
(^"  _  ;,')(&"  _  5x')  +  (y"  -  y'){Sy"  -  Sy')  +  {z"  -  z'){Sz"  -  Sz')  ^  ^  ^ 

^  7 

(x----.T-0(^x--^-^:g")  +  (/"-y")(^i/"^-VO+(^"^-^")(^^"^-fe"    ..  ^ 

These  being  multiplied  respectively  by  X'  and  X'",  and  added  to 
Equation  (526),  we  obtain  by  reduction,  and  by  the  principle  of 
indeterminate   co-efficients,  exactly  as  in  §213, 


X'-X'.^— r^=0; 


/ 


Y'  -  X 


,  y"-y' 


f 


z"  —  z 


(529) 


.'"        /»" 


X"  +  X' 
Y"  +  X 


/  9 


/ 


Z"  +  X' 


/ 


z'"  —  z" 
9 


(530) 


X'"  -\-X"'-- ^=0; 


Y"  +  X 


Z'"  +  X' 


nr      ^ J__  A  . 


9 


0; 


(531) 


=  0; 


Taking  from   each   group  its  first   equation  and  adding,  and   doing 
the  same  for   the  second   and   third,  we   have 


X'  +  X"  +  X'"  =  0  ; 
Y'  -f  Y"  +  Y'"  =  0 ; 
Z'  +  Z"  +  Z'"  =  0. 


(532) 


APPLICATIOXS. 


371 


That  is,   the   conditions   of   equilibrium  of    the  forces   are,   §80,   the 
same   as   though   they  had   been   applied  to   a   single   point. 

To  find  the  position  of  the  points,  eliminate  the  factors  X'  and 
X'",  and  for  this  purpose  add  the  first,  second  and  third  equations 
of  group  (530)  to  the  corresponding  equations  of  group  (531),  and 
there  will   result 

X"  +  X'"  +  y  (x"  -  x')  =  0  ; 
F"  +  r"'  +  y  (/'-y')  =  0; 
Z"  +  Z'"  +  ~  {z"  -  z')  =  0. 
from  which  we  find  by  elimination, 


Y"  +  V"  -  y'l  _  y\  {X"  +  X'")  =  0  ;  1 

z"  +  z'"  -  -C-^'  {X"  +  X'")  =  0.    I 

X     —  X  '  J 

From   group  (529),   by  eliminating  X', 


(533) 


X      —  X 


Z'  - 


ir  ^'  =  0  ; 


(534) 


and   finally  from   group   (531)  we   obtain,  by   eliminating   X'", 


y'"  -  y 

x'"  -  x' 


Z'"  — 


X'"  =  0 


X'"  =  0. 


(535) 


Equations  (532),  (533),  (534)  and  (535),  involve  all  the  conditions 
necessary  to  the  equilibrium,  and  the  last  three  groups,  in  connection 
with  group  (527),  determine  the  positions  of  the  points  A',  A" 
and  A"\  in  space. 


316. — The   reactions   in   the    system   which   impose  conditions  on 


372         ELEMENTS    OF    ANALYTICAL    MECHANICS. 

the   displacement  will    be    made    known   by   Equation   (331),   which 
because 

[^(;^^"7)J  "^[^(^''-y')]  ■^L^F^'^J  "^' 
\d{x"'-x")\  ^[d{y"'-y")\  ^\_d{z"'-z")\  -  '' 

becomes  for  the   cord  A'  A", 

X'  =  N' ; 
and  for  the  cord  A"  A'", 

\"'  =  N'"  ; 

from  which  we   conclude,  that   X'  and   X'"   are  respectively  the   ten- 
sions of  the   cords  A'  A"  and  A"  A'". 

This   is   also    manifest  from   Equations  (529)    and   (531);    for,  by 
transposing,  squaring,  adding  and   reducing   by  the   relations, 

{x"  -  x'Y  +  {y"  -  y'f  +  {z"  -  z'Y    _ 

P 

f^:,'"  _  :c"Y  +  {y'"  -  y"f  +  {z"'  -  z"Y  _ 

-^  _1, 

we  have 


X'    =.  -v/.Y'2    +  ¥'■'    +  Z'-'    =  R', 


\"'  =  -^x""'  +  y'"2  +  z"'2  =  R'", 


(536) 


in  which  R'    and  R'"  are   the    resultants  of    the   forces   acting   upon 
the  points  A'  and  A'"  respectively. 

Substituting   these  values   in   Equations  (529)  and  (531),  we  have 

X'        x"  -  x'        Y'        y"  -  y'       Z'        z"  -  z' 


R'  ~        f        '     R'  ^  f       '     R'  ~        f       ' 

X'"            x'"  -  x"      T"  _  y'"  —  y"  ^    Z'"  _  _  z'"  -  z"  ^ 

W'"^              ~g        '    'W'  ~  ~g        ''   W'  ~              'g         ' 

whence  the  resultants  of  the  forces  applied  at  the  points  A'  and  A"\ 
act  in    the   directions   of  the    cords    connecting   these  points  M^ith  the 

point  A",   and    will  be    equal    to,  indeed    determine  the   tensions    of 
these   cords. 


APPLICATIONS.  373 

§317. — From    Equations    (532),    wc    have    by    transposition, 

X"  =  -  {X'"  +  A'') ;    Y"  =  -  {¥'"  +  V)  ■    Z"  =  -  (Z'"  +  Z'). 

Squaring,    adding   and    denoting    the    resultant   of    the   forces   applied 
at  A"  by  R'\  we  have 

E"  =  ^{X'"  +  A")"  +  {V'"  +  ry-  +  [Z'"  +  z')-  •  •  (r>37) 

and    dividing  each  of  the   above    equations   by  this  one 

X'"  +  X'     1 


jr 

E" 

Y" 
R" 


R" 

Y'"  +  Y' 
R" 


(538) 


<?^ 


^^/>/  ^  '^z  "  -^^'^^      -^ 

lelosram  ^"  w  C«  be  constructed, 
.4"  (7  will  represent  the  value  of 
R".  If  A'  A"  A'"  be  a  contin- 
uous cord,  and  the  point  A" 
capable  of  sliding  thereon,  the 
tension  of  the  cord  would  be 
the  same  throughout,  in  which 
case  R'  would  be  equal  to  R"\ 
and  the  direction  of  R"  would 
bisect  the  angle  A'  A"  A'". 

The  same  result  is  shown  if, 
instead  of  making  ^/  =  0  and 
5^  =  0     separately,     we     make 


A"' 


372  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

the   displacement   will    be    made    known   by   Equation   (331),   which 
because 

[^^^TTz-^J  +[rfT7^"7)]  ■^L^F^'^J  "^' 

\_d{x"'-x")\   +  [d{y"'  -  y")]   ^  Id  (.'"  -  .")J   ~      ' 
becomes  for  the   cord  A'  A", 

■k'  =  JV'; 
and  for  the  cord  A"  A'", 

X'"  =  N'"  ; 

from  which  we   conclude,  that   X'  and   X'"   are  respectively  the   ten- 
sions of  the   cords  A'  A"  and  A"  A'". 


X'"  =  yx'"2  _^  Y"'2  _|_  z'"^  =  B'",  ) 


\ya\j) 


in  which  B'  and  B'"  are  the  resultants  of  the  forces  acting  upon 
the  points  A'  and  A'"  respectively. 

Substituting   these  values   in   Equations  (529)  and  (531),  we  have 

^  _  ''"  -  ""'        H  -  ^"  ~  y' '     —  =  ^"  ~  ^'  ' 
R  -        f        '     R'  "        f       '     B'  ~        /' 

X'"  x'"  —  x"      Y'"  y'"  —  y"      Z'"  _       z'"  —  z"  ^ 

whence  the  resultants  of  the  forces  applied  at  the  points  A'  and  A'", 
act  in  the  directions  of  the  cords  connecting  these  points  with  the 
point  A",  and  will  be  equal  to,  indeed  determine  the  tensions  of 
these   cords. 


APPLICATIONS.  373 

§317. — From    Equations    (532),    \\c    have    by    transposition, 

X"  =  -  (A'"'  +  A") ;    Y"  =  -  {V"  +  V) ;    Z"  =  -  (Z'"  +  Z'). 

Squaring,    adding    and    denoting    the    resultant   of    the    forces   applied 
at  A"  by  R",  we  liuve 


E"  =  -^{X"'  -f-  Xy  +  {V'"  +  F'f  +  (Z'"  -f  Z')-'  •  •  (537) 
and    dividing  each   of  the   above    equations   by  this  one 


X" 

X'"-  + 

X' 

R" 

R" 

Y" 

Y'"  + 

Y' 

R" 

R" 

Z" 
R" 

= 

Z' 

'  + 
R" 

Z' 

(538) 


4", 


whence,  Equation  (537),  the  resultant  of  the  forces  applied  at  A"  is 
equal  and  immediately  opposed  to  the  resultant  of  all  the  forces 
applied   both  at  A'  and  ui"' 

If,  therefore,  from  the  point 
A'\  distances  A"  m  and  A"  n 
be  taken  proportional  to  R'  and 
R'"  respectively,  and  a  paral- 
lelogram A"  m  Cn  be  constructed, 
A"  C  will  represent  the  value  of 
R".  If  A' A"  A'"  be  a  contin- 
uous cord,  and  the  point  A" 
capable  of  sliding  thereon,  the 
tension  of  the  cord  would  be 
the  same  throughout,  in  which 
case  R'  would  be  equal  to  R"\ 
and  the  direction  of  R"  would 
bisect  the  angle  A'  A"  A'". 

The  same  result  is  shown  if, 
instead  of  making  iJ/  =  0  and 
5  <7  =  0     separately,     wc     make 


A"' 


374         ELEMENTS     OF    ANALYTICAL    MECHANICS. 


^  (y  ^  ^^  =:  0,    multiply    by   a     single     indeterminate     quantity    X, 
and  proceed    as    before. 

§  318. — Had  there  been  four 
points,  A',  A",  A'",  and  A^\ 
connected  by  the  same  means, 
the  general  equation  of  equili- 
brium would  become,  by  call- 
ing h  the  distance  between  the 
points,  A'"  and  ^'^, 

X'  §  x'  -f-  X"  S  x"  -f  X'"  S  x'"  +  X'^^  S  x^^ 
-f  Y'  8y'  +  Y"  S  y"  +  Y"  S  y'"  +  Y'^  S  yiv 
+  Z'  5  z'  4-  Z"  8  z"  +  Z'"  S  z'"  +  Z''  5  2*^ 
+  X'  Bf  A-  >."  Sp    4-  'k'"  S  h 

and   from  which,  by  substituting    the  values  of  S  f,  5  g,  and  S  h,  the 
following  equations  will  result,  viz. : 


[   =0; 


X'  -X'  - 
F'  —  X'  . 
Z'  —  X' . 


/ 

f 

z"  -  z' 


.  f 


0, 
0, 
0, 


(539) 


3."    —    x'  X 

X'4X'.^--x" 

z"    —   z' 
Z"  +-K   >  ? -  X' 


/ 


X'"  4-  X" 


x'"  —  X' 


Y"  4-  X"  .  ^ ^^^^  -  X'"  .  ^ i-^-  =  0,    } 


Z'"  4-  X"  . X'" 


(540) 


(541) 


APPLICATIONS. 


3-ir   X 

Xiv  +  X'" ^ =  0, 


Y'^  +  X 


,r>  f-y 


=  0, 


2.V    +  X'" =  0, 


(542) 


Eliminating  the  indeterminate  quantities  X[  X",  and  X'",  we  obtain 
nine  equations,  from  which,  and  the  three  equations  of  conditions 
expressive  of  the  lengths  of/,  ff,  and  h,  the  position  of  the  points  A', 
A",  A'",  and  A^"  may  be  determined. 

If  there  be  n  points,  connected  in  the  same  way  and  acted  upon 
by  any  forces,  the  law  which  is  manifest  in  the  formation  of  Equa- 
tions (539),  (540),  (541),  and  (542),  plainly  indicates  the"  following 
n  equations  of  equilibrium : 


X'  -  X' .  ^—^  =  0, 
/ 

p  _  X'  .  ^^-^  =  0, 


z'  -  X' 


/ 

x"  -  z' 

f 


=  0, 


(543) 


X"  +  X' 


/ 


X"  .    ^-=-^-   ::.   0, 

9   ■ 


Z"  +  X'  .  ^— ^  -  X' 


jn  ~n 


=  0, 


(544) 


III    _       II  ^iv    _   x'"  ^ 

X'"  +  X"  .  ^ —  -  X'"  . =  0, 


h 


Y'"  +  X 


Z'"  +  X" 


iij"'-y"  _yii.yZjz:j::^o, 


9 

.III        ~" 


-  X'" 


2"    —    z' 


0, 


(545) 


376  ELEMENTS     OF    ANALYTICAL    MECHANICS, 


^„-i  +  K- 


^„-i    +  K- 


a!n-i 

—  ar„_2 

k 

y.-i 

—    2/.-2 

k 

2,-1 

—  2;„_2 

—  X. 


—  X, 


Z„  +X„_,.?^i^l^  =  0. 


—  ^„-i 

/ 

y» 

—  y„-i 

z 

2„ 

—  2„-i 

=  0, 


=  0, 


(54G) 


(547) 


In  which  X,  with  its  particular  accent,  denotes  the  tension  of  the 
cord  into  the  difference  of  whose  extreme  cordinates  it  is  multi- 
plied. 

Adding  together   the    equations   containing   the    components  of  the 
forces  parallel   to  the  same  axis,   there  will  result 


X'  +  X"  +  X'"  +  X''  .  .  •  X„  =  0, 
Y'  +  Y"  +  Y'"  +  Y'"  •  •  •  F„  =  0, 
Z'   +  Z"  +  Z'"  +  Z'^      .     .     .     Z„  =  0,  J 


(548) 


from   which   we    infer,    that    the    conditions   of    equilibrium    are   the 
same  as  though  the  forces  were  all  applied  to  a  single  point. 

From   group  (543),  we   find  by  transposing,  squaring,  adding  and 
extracting    square   root. 


and  dividing  each  of  the  equations  found  after  transposing   in   group 
(543)  by  this   one. 


X' 

= 

rr" 

— 

x' 

R' 

/ 

) 

T 
'R' 

= 

^ 

y 

y' . 
> 

Z' 

= 

z" 

— 

z' 

R' 

f 

APPLICATIONS. 


377 


Treating    the    equations  of    group    (•J47)    in    the     same    way,    we 
have 


^n 

—   ^«-l 

I 

Vn 

-y„-i 

I 

Zn 

~n  — 1 

/ 


whence,  the  resultants 

of  the   forces   applied 

to  the  extreme  points 

A'  and  A^,  act  in  the 

direction  of  the  extreme  cords.     And  from  Equations  (548)  it  appears 

that   the   resultant   of  these    two   resultants   is  equal   and   contrary  to 

that    of  all    the    forces   applied   to    the    other   points. 

§319.— If  the  extreme  points  be  fixed,  X\  V,  Z'  and  X„,  F„,  Z„, 
will  be  the  components  of  the  resistances  of  these  points  in  the 
directions  of  the  axes ;  these  resistances  will  be  equal  to  the  ten- 
sions X^  and  X„  of  the  cords  which  terminate  in  them.  Taking  the 
sum  of  the  equations  in  groups  (543)  to  (547),  stopping  at  the  point 
whose   co-ordinates   are  a:„_,„,  y„_„, ,  2„_,„,  we   have 


X'  +  ^X-  K 

y  +  2  r  _  X,, 


a*,,-™  —  x„ 


=  0 


y„-„ 


y„-. 


±  =  0 


z'  +  ^z  -  x„_,.^ -^"^^ 


i  =  0 


(549) 


in  which  2  X,  2  Y^  2  Z,  denote  the  algebraic  sums  of  the  components 
in  the  directions  of  the  axes  of  the  active  forces ;  X,_„,.jthc  tension 
on  the  side  of  which  the  extreme  co-ordinates  are  a;„_„,  y,_„,  «,_« , 
and  x„_„_i,  y„-m-i)  ~n-m-i;   and  /„_„  the  length  of  this   side. 


s  320 — Now,    suppose   the    length   of    the    sides    diminished    and 


37S         ELEMENTS     OF    ANALYTICAL    MECHANICS. 


their    number    increased     indefinitely ;    the    polygon    will     become    a 
curve  ;    also,   making   X^„  =  t,  w^   have 

a^,^m    —    Xn~m-l    =    d  X, 

Vn-m  —  yn-,,^1  =  dy, 

z.-M.  —  2»-m-i  -  dz, 

/„_„  =  ds, 

s  being   any  length  of  the   curve  ;  and  Equations  (549)  become 

dv 
X'  +  2  X  -  f  •  —  =  0  ; 
a  s 


dy 


Y'  +  '^Y  -  t--r  =  0  ; 
ds 


Z'  +  :eZ  -  t--f  =  0; 
d  s 


(550) 


which  -will  give  the  curved  locus  of  a  rope  or  chain,  fastened  at 
its  ends,  and  acted  upon  by  any  forces  whatever,  as  its  own  weight, 
the  weight  of  other  materials,  the  pressure  of  winds,  currents  of 
water,  &c.,  &c. 

This  arrangement  of  several  points,  connected  by  means  of  flexi- 
ble cords,  and  subjected  to  the  action  of  forces,  is  called  a  Funi- 
cular Machine. 

§321. — If  the   only    forces   acting   be    pressure   from   weights,   we 

have,  by  taking   the   axis   of  z  vertical, 

X,,, 

X"  =  X'"  =  A'i^  &c.^":L  0 ;     Y"  =  Y'"  &c.  =  0 ; 

and    from   Equations  (543)   to   (54T), 

r"  _  rr'  x'"  —  x" 

JiT'^X'.^-^  =X".^ ^  =    •  . 

/  9 


whence,  the  tensions  on  all  the  cords,  estimated  in  a  horizontal 
direction,  are  equal  to  one  another.  Moreover,  we  obtain  from  the 
same    equations,    by  division, 


y"  -  y'  _  y'"  -  v"  ^ 

a;"  —  x'~   x'"  -  x" 


Xn  —  a:„_i 


APPLICATIONS. 


379 


These  are  the  tangents  of  the  angles  which  the  projections  of  the 
sides  on  the  plane  xy  make  wifli  the  axis  x.  The  polygon  is 
therefore   contained  in   a   vertical   plane. 


THE   CATENARY. 


8  322. — If  a  single  rope  or  chain  cable  be  taken,  and  subjected 
only  to  the  action  of  its  own  weight,  it  will  assume  a  curvilinear 
shape  called  the  Catenary  curve.  It  will  lie  in  a  vertical  plane. 
Take  the  axes  z  and  x  in  this  plane,  and  z  positive  upwards,  then 
will 

2X=0;     2F=:0;     F'  =  0;     :l  Z  =  -  W ; 

in   which   W  denotes   the  weight  of  the   cable,  and    Equations  (550) 

become 

dx 


X'  -t 


ds 
dz 


=  0, 


Z'  -W  -t-—  =  0. 
a  s 


(551] 


These    are    the    differential    equations   of   the   curve.     The    origin 
may  be   taken  at  any  point. 
Let  it  be  at  the  bottom  point  2 

of  the  curve.  The  curve 
being  at  rest,  will  not  be 
disturbed  by  taking  any  one 
of  its  points  fixed  at  pleas- 
ure. Suppose  the  lowest 
point  for  a  moment  to  be- 
come  fixed.      As   the    curve 

is  here  horizontal,  Z,  =  0,  §319,  and  from   the   second  of  Equations 
(551),  we   have 

dz 


W  = 


ds' 


(552) 


whence,  the  vertical  component  of  the  tension  at  any  point  as  0  of 
the  curve,  is  equal  to  the  weight  of  that  part  of  the  cable  between 
this  point  and  the  lowest  point.     The  first  of  Equations  (551)  shows 


380  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

that  the  horizontal  component  of  the  tension  at  0  is  equal  to  the 
tension  at  the  lowest  point,  as  it  should  be,  since  the  horizontal 
tensions   are  equal  throughout. 

Taking  the  unit  of  length  of  the  cable  to  give  a  unit  of  weight, 
which  would  give  the  common  catenary,  we  have  W  =  s ;  and,  de 
noting    the   tension   at    the   lowest   point  by  r,  we  have 

and  from  Equation  (552), 

s''  d  s 
dz  =z  rzp  —  • 


Taking  the  positive  sign,  because  z  and  s  increase  together,  inte- 
grating, and  finding  the  constant  of  integration  such  that  when 
2  =  0,  we   have   5  =  0, 

z  +  c  =  /^M^ ; 
whence, 

S2    =   z2   _|_    2c2. 

Also,  dividing  the  first    of  Equations  (551)  by  Equation  (552), 

dx  c  c 

dz   ~     S     ~     y^22  ^  2cz  ' 

and  integrating,  and  taking  the  constant  such  that  x  and  z  vanish 
together, 


z  +  c  +  ^z'  +  2cz 

X  =  c  '  log •      •      •     (oo3) 

c 

which   is    the    equation  of  the   catenary. 

This   equation   may   be   put    under    another   form.      For  we   may 
write    the   above, 


ce7=z-\-  c  +  -y/{z  +  c)  2  :i-  c2 ; 
transposing  z  -{-  c  and   squaring, 

c2  .  e"r  _  2  c  e^  (2  +  c)  =  —  c^ ; 
whence, 

X  X 

2  +  c  =  I  c  .  (e^  +  e~7). (554) 


APPLICATIONS.  3S1 

Also, 

and   by   substitution, 

z  X 

s  =  \c'[y  —  e~~). (555) 

§323. — If  the  length  of  the  portion  of  the  cable  which  gives  a 
unit  of  weight  were  to  vary,  the  variation  might  be  made  such  as 
to  cause  the  area  of  the  cross  section  to  be  proportional  to  the 
tension  at  the  point  where  the  section  is  made.  The  general  Equa- 
tions (551)  will   give   the   solution   for   every  possible  case. 

/  d  <f«^  ; —  S  frO 


i^e/- 


L 


■{.CH-^  U/^~~Y.  t^C-hlcxj 


CC>^-^1_CU^         ^/'^rr^i  ^  ,  „       ^ 


1 


If    .      tlI\P       OWV^llVI.     **  v^ 


Denoting  by  d  the  angle  abt,^ 
and  by  p  the  resultant  &  m  of 
these  forces,  which  is  obviously 
the    pressure  of  d  s  against  the   cylinder,  we   have.  Equation  (5G), 


2)  z=  y/C~  +  /-  4-  2  / .  ^  cos  ()  =  t  -y/2(l  +  cos(?)  ; 
but 

1  +  cos  d  =  2  \i4l?  i  d  ;      (180°  -  C)  =  —  ; 


380  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

that  the  horizontal  component  of  the  tension  at  0  is  equal  to  the 
tension  at  the  lowest  point,  as  it  should  be,  since  the  horizontal 
tensions   are   equal  throughout. 

Taking  the  unit  of  length  of  the  .cable  to  give  a  unit  of  weight, 
which  would  give  the  common  catenary,  we  have  W  =  s ;  and,  de 
noting   the   tension  at   the  lowest   point  by  c,  we  have 

and  from  Equation  (552), 

s'-  d  s 


This   equation   may   be   put    under    another   form.      For  ^ye   may 
write    the   above, 


transposing  z  -{-  c  and    squaring, 

c2  .  e"—  _  2  c  e"^  {z  +  r)  =  —  c^  ; 
whence, 

z  +  c  =:  I  c  •  (e^  +  e~~^). (554) 


AT  PLICATIONS. 


3S1 


Also, 

and   by   substitution, 


=  V(2  +  cf  -  c\ 


s  =  ^C'  {€'  —  e    '). (555) 

1 323. — If  the  length  of  the  portion  of  the  cable  Avhich  gives  a 
unit  of  weight  were  to  vary,  the  variation  might  be  made  such  as 
to  cause  the  area  of  the  cross  section  to  be  proportional  to  the 
tension  at  the  point  where  the  section  is  made.  The  general  Equa- 
tions (551)  will   give   the   solution   for   every  possible  case. 


FKICnON   BET^VEEN   CORDS   AND   CYLINDEICAL   SOLIDS. 

1 324. — ^V'hen  a  cord  is  wrapped  around  a  solid  cylinder,  and 
motion  is  communicated  by  applying  the  power  F  at  one  end 
while  a  resistance  W  acts  at  the  other,  a  pressure  is  exerted  by 
the  cord  upon  the  cylinder ;  this  pressure  produces  friction,  and  this 
acts  as  a  resistance.  To  estimate  its  amount,  denote  the  radius 
of  the  cylinder  by  i2,  the  arc  of  contact  by  s,  the  tension  of  the 
cord    at   any  point  by  t. 

The  tension  t  being  the  same 
throughout  the  length  ds  =  at^ 
of  the  cord,  this  element  will  be 
pressed  against  the  cylinder  by 
two  forces  each  equal  to  /,  and 
applied  at  its  extremities  a  and  t^ , 
the  first  acting  from  a  towards 
W.  the  second  from  t^  towards  b'. 
Denoting  by  d  the  angle  ab  t^^ 
and  by  -p  the  resultant  b  m  of 
these  forces,  which  is  obviously 
the   pressure  of  ds  against  the   cylinder,  we   have.  Equation  (50), 

2-)  =  ^/fi  +  t~  -{-  '2t~ll^d  =  t  -y/i  (I  -f-  cost?)  ; 
but 

"^^^  d  s 

1  +  cos  ^  =  2  Klll^  ^  ^  ;     (ISO*^  -  ^)  =  —  •, 


382  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and   taking   the   arc   for    its    sine,    because    180°  —  ^   is   very   small, 
■we  have 

ds 

and   hence,  §308,  the  friction  of  c?s  will  be 

The  element  t^  t^  of  the  cord  which  next  succeeds  a  t, ,  will  have 
its  tension  increased  by  this  friction  before  the  latter  can  be  over- 
come ;  this  friction  is  therefore  the  differential  of  the  tension,  being 
the   difference   of  the   tensions  of  two  consecutive  elements ;  whence, 

dt=f-t--; 
dividing   by    t   and   integrating, 

log^=/-^  +  logC, 
or, 

t  z=  Ce'^ (556) 

making  s  =  0,  we   have  t  =  W  =   C ;  whence, 

t  =  W-  e^; (557) 

and   making  s  =  >S  =  a^i  4  fa,  we   have  t  —  F\  and 

fa 
F—W-eR       (558) 

Suppose,  for  example,  the  cord  to  be  wound  around  the  cylinder 
three  times,  and  /  r=  l  ;  then  will 

^  =  3*  .  2  i2  =  6  .  3,1416  .  i2  =  18,849i?, 

and 

i.=  Trx.^^"'^^^=Trx  (2,71825)^'""; 

or, 

F=  TF.  535,3; 

that  is   to   say,  one   man   at  the  end   W  could   resist  the    combined 
effort  of  535   men,  of  the  same   strength  as  himself,,  to   put  the  cord 


APPLICATIONS. 


3S3 


in  motion  -svlien  wound  three  times  around  the  cylinder.  This  explains 
why  it  is  that  a  single  man,  by  a  few  turns  of  her  hawser  around  a 
dock-post,  is  enabled  to  prevent  the  progress  of  a  steamboat  although 
her  machinery  may  be  in  motion.  Here  friction  comes  in  aid  of 
the  power,  and  there  are  numerous  instances  of  this ;  indeed,  with- 
out friction,  many  of  the  most  useful  contrivances  and  constructions 
would  be  useless.  It  is  by  the  aid  of  friction  that  the  capstan  is 
enabled  to  do  its  work ;  the  friction  between  the  rails  of  a  rail- 
road and  the  wheels  of  the  locomotive  enables  the  latter  to  put 
itself  and  its  train  of  cars  in  motion.  But  for  the  friction  between 
the  feet  of  draft  animals  and  the  ground,  they  could  perform  no 
work  ;  nor,  indeed,  could  any  animal  walk  or  even  stand  with  safety, 
if  they  were  deprived  of  the  aid  of  this  principle. 


IXCLIN'ED    PLANE. 

§  325. — Tlie  inclined  plane  is  used   to  support,  in  part,  the  weight 
of  a  body  while  at  rest  or  in  motion  upon  its  surface. 

Let  any  body  M,  rest  with  one  of  its  faces  in  contact  with  the 
inclined  plane  A  B.  De- 
note its  weight  by  IF,  and 
suppose  it  to  be  solicited 
by  a  force  F  in  the  direc- 
tion G  Q,  making  with  the 
inclined  plane  the  angle 
QGq\  which  denote  by  (p. 
Denote  the  inclination  B  AC 
of  the  plane  to  the  horizon 
by  a.  Resolve  the  woigiit 
W  =^  G  G'  into  two  com- 
ponents, Gp  and  Gp',  one 
perpendicular  and  the  other 
parallel  to   the  plane.     The 

angle    G'  G  p   being    equal   to   the    angle   B  A  (7,   the   first   of   these 
components  will  be, 

G  p  =  W  .  cos  a  ; 


384  ELEMENTS     OF     ANALYTICAL    MECHANICS. 

and  the  second, 

Op'  =   W  .  sin  a. 

In  like  manner,  resolve  the  force  F  =  G  Q  into  two  components 
Gq  and  Gq',  the  first  normal  and  the  second  parallel  to  the  plane. 
The  first  of  these  will  be, 

Gq  ~  F .  sin  (p  ; 
and  the  second, 

G  q'  =  F .  cos  (p. 

The  total  pressure  upon  the  plane  will  be, 

W  .  cos  a  —  i^ .  sin  9 ; 
and  the  friction  thence  arising, 

/  (  PT  .  cos  a  —  i>^ .  sin  (p)  ; 

in  which  /  denotes  the  coefficient  of  friction.  The  force  which 
solicits   the   body  in    the   direction    vp   the   plane   is 

F .  cos  9  ; 
whence,  Equation  (507), 

I.  P8p  =  i^ .  cos  (p  .  (?  s  ; 
2  Q  8  q  —  f  {W  cos  a  —  F  s\\\  (];>)  d  s  -^^   W  sin  ads, 

in  which  J  s  is  the  elementary  path  described  on  the  plane ;  and  when 
the  body  is  moving  uniformly,  wc  have.  Equation  (508), 

F  cos  (^  d  s  —  f{W  cos  OL  —  F .  sin  (p)  c?  5  —  IF  sin  a  c?  s  =  0  ; 

whence, 

^  ^    Tr(sina+/cosa) ^^.^^ 

cos  9  +  /  sin  9 

This  supposes  motion  to  take  place  up  the  plane;  if  the  power  F 
be  just  sufficient  to  permit  the  body  to  move  uniformly  down  the 
plane,  then  will  /  change  its  sign,  and  we  shall  have 

p  ^   W{sma-fcosa)^ ^^^^^ 

cos  9  —  /  sin  9 

And  the  power  may  vary  between  the  limits  given  by  these  two 
values  without  moving  the  body. 


APPLICATIONS.  3S5 

§  326.  If  the  power  be  zero,  or  i^  =  0,  then  will 

sin  a  —  /  cos  a  =  0, 
or 

tan  a  =  /, 

which  is  the  angle  of  friction,  §    308. 

g  307^ — jf  the    power   act   parallel   to  the  plane,  then  will  9  =  0, 

and 

i^z=  Tr(sina  ±/cosa) (5G1) 

the  upper  sign  answering  to  the   case  of  motion   uj),  and   the   lower, 
down  the  plane ;    the  difference  of  the  two  values  being 

2 /cos  a. 
If  /  =  0,  then   will 


F 

BC 



— 

sin 

a 

W 

AB 

that  is,  the  power  is  to  the  weight  as  the  height  of  the  plane  is  to 
its  length. 

§  328. — If  the  power  be  applied  horizontally,  then  will  9  be  nega- 
tive and  equal  to  a,  and  we  have,  by  including  the  motion  m  both 
directions, 


F  BC 

-^=tana  =  — . 

That   is,  the   power  will    be   to    the    resistance    as   the  height  of  the 
plane  is  to  its  base. 

^329. — To  find  under  what  angle  the  power  will  act  to  greatest 
advantage,  make  the  denominator  in  Equation  (559)  a  maximum. 
For  this  purpose,  we  have,  by  differentiating,  ^ef""-^'"i'e  tixt^ 

—  sin  9  +  /cos  9  =  0; 
25 


384  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and  the  second, 

G  -p'  rz:   W  .  sin  a. 

In  like  manner,  resolve  the  force  F  =  G  Q  into  two  components 
G  q  and  G  q',  the  first  normal  and  the  second  parallel  to  the  plane. 
The  first  of  these  will  be, 

Gq  =  F .  sin  cp ; 
and  the  second, 

G  q'  =  F  .  cos  9. 

The  total  pressure  upon  the  plane  will  be, 

W  .  cos  a  —  i^ .  sin  (p ; 

and  the  friction  thence  arising, 

/  (  PT  .  cos  a  —  i^ .  sin  9)  ; 

in  which  /  denotes  the  coefficient  of  friction.  The  force  M'hich 
solicits   the   body  in    the   direction    vji   the   plane   is 

F .  cos  (p  ; 
whence,  Equation  (507), 

^  FSp  =  F .  cos  (p  .  ds; 

:^/±: 


~  COS  9  +  /  sin  9 

This  supposes  motion  to  take  place  up  the  plane  ;  if  the  power  F 
be  just  sufficient  to  permit  the  body  to  move  uniformly  down  the 
plane,  then  will  /  change  its  sign,  and  Ave  shall  have 

jr  ^   ^(sina-/cosa)^ ^^^^^ 

cos  9  —  /  sin  9 

And  the  power  may  vary  between  the  limits  given  by  these  two 
values  without  moving  the  body. 


APPLICATIONS.  3S5 

§  326.  If  the  power  be  zero,  or  i^  =  0,  then  will 

sin  a  —  /  cos  a  =  0, 
or 

tan  a  =z  /, 

which  is  the  angle  of  friction,  §    308. 

g  327^ — If  the    power   act   parallel    to  the  plane,  then  will  9  =  0, 

and 

i^=:  Tr(sina  ±/cosa) (.501) 

the  upper  sign  answering  to  the   case  of  motion   up,  and   the   lower, 
down  the  plane ;    the  difference  of  the  two  values  being 

2 /cos  a. 
If  /  =  0,  then   will 

F         .  B  C 

that  is,  the  power  is  to  the  weight  as  the  height  of  the  plane  is  to 
its  length. 

§  328. — If  the  power  be  applied  horizontally,  then  will   (p  be  nega- 
tive and  equal  to  a,  and  we  have,  by  including   the   motion   in  both 

directions, 

^^   Tr(sin«^±/cos_a) 

cos  a  q:  y  sin  a  ^        ' 

the  difference  of  the  limiting  values  being 

2/.  W 


cos^  a.  —  f^  sin^  (p  ■:' 
If  the  friction  be  zero,  or  /  =  0,  then  will 

F  BC 

-=tana  =  — . 

That   is,  the   power  will    be   to    the    resistance    as   the  height  of  the 
plane  is  to  its  base. 

§329. — To  find  under  what  angle  the   power  will  act   to   greatest 
advantage,  make    the   denominator    in    Equation   (559)    a   maximum. 


For  this  purpose,  we  have,  by  differentiating,  ^  e  7}  i^y. .  3  >  >>  r  e  n't?, 

—  sin  ip  +  /cos  9  =  0; 
25 


386 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


whence, 

tan  (p  =  /. 

That  is,  the  angle  should  be  positive,  and  equal  to  that  of  the  fric- 
tion. 

§330. — If  the  power  act  parallel  to  any  inclined  surface  to  move 
a  body  up,  the  elementary  quantity  of  work  of  the  power  and  resist- 
ances will  give  the  relation,  Equation  (561), 

F d  s  =  TFc?ssina  -j-  Wf  d  s  cos  a. 

But,  denoting  the  whole  hori- 
zontal distance  passed  over  by 
1  =  A  C,  and  the  vertical  height 
by  h  =  £  C,  we  have 

d  s  .  sin  a  =  d  h, 

d  s  .  cos  a  :=  d  I; 

whence,  substituting,  and  integrating,  and  supposing  the  body  to  be 
started  from  reet  and  brought  to  rest  again,  in  which  case  the  work 
of  inertia    will    balance  itself,  we  have 

Fs=Wk+f.W.l, (563) 

in  which  there  is  no  trace  of  the  path  actually  passed  over  by  the 
body.  The  work  is  that  required  to  raise  the  body  through  a  ver- 
tical height  B  C,  and  to  overcome  the  friction  due  to  its  weight  over 
a   horizontal    distance   A  C. 

The  resultant  of  the  weight  and  the  power  must  intersect  the 
inclined  plane  within  the  polygon,  formed  by  joining  the  points  of 
contact  of  the  body,  else  the  body  will  roll,  and   not  slide. 


THE    LEVEK. 


§331. — The  Lever  is  a  solid 
bar  A  B,  of  any  form,  supported 
by  a  fixed  point  0,  about  which 
it  may  freely  turn,  called  the  ful- 
crum. Sometimes  it  is  supported 
upon     trunnions,    and    frequently 


APPLICATIONS. 


387 


upon  a  knife-edge.  Levers  have 
been  divided  into  three  different 
classes,  called  orders. 

In  levers  of  the  first  order,  the 
power  F  and  resistance  Q  are 
applied  on  opposite  sides  of  the 
fulcrum  0;  in  levers  of  the  second 
order,  the  resistance  Q  is  applied 
to  some  point  between  the  ful- 
crum 0  and  the  point  of  appli- 
cation of  the  power  F ;  and  in 
the  third  order  of  levers,  the 
power  F  is  applied  between  the 
fulcrum  0  and  point  of  applica- 
tion of  the  resistance   Q. 

The  common  shears  furnishes 
an  example  of  a  pair  of  levers 
of  the  first  order ;  the  nut-crackers 
of  the  second ;  and  firc-tongs  of 
the  third.  In  all  orders,  the  con- 
ditions of  equilibrium  are  the 
same. 

These  divisions  are  wholly  ar- 
bitrary, being  founded  in  no  dif- 
ference of  principle.  The  relation 
of  the  power  to  the  resistances, 
is   the  same    in  all. 

Let  A  B  be  a  lever  supported 
upon  a  trunnion  at  0,  and  acted 
upon  by  the  power  P  and  resist- 
ance Q,  applied  in  a  plane  per- 
pendicular to  the  axis  of  the  trun- 
nion. Draw  from  the  axis  of  the 
trunnion,  the  lever  arms  On  and 
0  ni,  being  the  perpendicular  dis- 
tances of  the  power  and  resistance 
from     the    axis    of    motion,    and 


388  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

denote   them   respectively  by  l^  and  l^ ;  also    denote  the   resultant  of 
P  and   Q  by  iV,  the   radius  of  the   trunnion   by  r,  the  co-efficient  of 
friction   by  /,  and   the    arc   described   at  the  unit's   distance  from    the 
axis  by  Si . 
Then, 

dp  =  Ij, .  d  Si  ',     d  q  =z  l^ .  d  Si, 

N=  -^F^  +  Q^  +  2P  Qcosd, 

in  which  6  is  the  angle  of  inclination  A  C  B  of  the  power  to  the 
resistance.  Then,  supposing  the  lever  to  have  attained  a  uniform 
motion,  will,  Equations  (508)  and  (524), 

P.l^.ds,  -Q.K.ds,  -y^P^+  Q^  +  2FQcos6-  Jll^A^  ^  0.(564) 

Omitting  the  common  factor  d  s, ,  and  making 

f  .  ^r  r 

we  have, 

P  -mQ  -  ^P^  +  §2  +  2  P  §  .  cos  ^  -fn  =  0. 
Transposing,  squaring,  and  solving,  with  respect  to  P,  we  find. 


If  the  fraction  n  be  so  small  as  to  justify  the  omission  of  every 
term  into  which  it  enters  as  a  factor,  or  if  the  co-efficient  of  friction 
be  sensibly  zero,  then  would 

l="'=i ^''''> 

That  is,  the  power  and  the  resistance  are  to  each  other  inversely  as 
the  lengths  of  their  respective  lever  arms. 

If  the  power  or  the  resistance,  or  both,  be  applied  in  a  plane 
oblique  to  the  axis  of  the  trunnion,  each  oblique  action  must  be 
replaced  by  its  components,  one  of  which  is  perpendicular,  and  the 
other  parallel  to  the  axis  of  the  trunnion.  The  perpendicular  com- 
ponents must  be  treated  as  above.     The  parallel  components  will,  if 


APPLICATIONS. 


389 


the  friction  arising  from  the  resultant  of  the  normal  components  be 
not  too  great,  give  motion  to  the  whole  body  of  the  lever  along  the 
trunnion  ;  and  if  this  be  prevented  by  a  shoulder,  the  friction  upon 
this  shoulder  becomes  an  additional  resistance,  whose  elementary 
quantity  of  work  may  be  computed  by  means  of  Eq.  (520)  and  made 
another  term  in  Equation  (564). 


WHEEL   AND   AXLE. 

§332. — This  machine  consists  of  a  wheel  mounted  upon  an  arbor, 
supported  at  either  end  by  a  trun- 
nion resting  in  a  box  or  trunnion 
bed.  The  plane  of  the  wheel  is  at 
right  angles  to  the  arbor  ;  the  pow- 
er P  is  applied  to  a  rope  wound 
round  the  wheel,  the  resistance  to 
another  rope  wound  in  the  opposite 
direction  about  the  arbor,  and  both 
act  in  planes  at  right  angles  to  the 
axis  of  motion.  Let  us  suppose  the 
arbor  to  be  horizontal  and  the  re- 
sistance §  to  be  a  weight. 

Make 
iV  and  N'  =  pressures  upon  the  trunnion  boxes  at  A  and  B; 
R  =  radius  of  the  wheel ; 
r  =  radius  of  the  arbor ; 
p  and  p'  =  radii  of  the  trunnions  at  A  and  B ; 


4 


Sx  =  arc   described    at    unit's  distance    from  axis  of  motion. 
Then,  the  system  being  retained  by  a  fixed  axis,  we  have 

P  S  p  =  PEd  s^; 

Q  d  q  =z  Q  r  d  Si. 

The  elementary  work  of  the  friction  will,  Eq.  (524),  be 


390  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and  the  elementary  work  of  the  stiffness  of  cordage,  Equations 
(515), 

d, — T-dsr, 

and  when  the  machine  is  moving  uniformly, 

The  pressures  N  and  N'  arise  from  the  action  of  the  power  P,  the 
weight  of  the  machine,  and  the  reaction  of  the  resistance  §,  in- 
creased by  the  stiffness  of  cordage.  To  find  their  values,  resolve 
each  of  these  forces  into  two  parallel  components  acting  in  planes 
which  are  perpendicular  to  the  axis  of  the  arbor  at  the  trunnion 
beds;  then  resolve  each  of  these  components  which  are  oblique  to 
the  components  of  Q  into  two  others,  one  parallel  and  the  other 
perpendicular  to  the  direction  of  Q. 

Make 
w   =  weight   of  the   wheel   and   axle, 
g    =  the    distance   of  its   centre    of  gravity  from  -4, 
p    =  the    distance  m  A, 
q    —  the   distance   n  A, 
I    =  length   of  the   arbor   A  B, 
9    =:  the   angle   which   the   direction   of  F  makes   with   the   vertical 

or    direction    of    the   resistance    Q. 
Then   the   force   applied   in   the   plane  perpendicular   to   the   trunnion 
A,  and  acting  parallel  to  the  resistance   Q,  will,  §  95,  be, 

and  the  force  applied  in  this  plane  and  acting  at  right  angles  to  the 
direction  of  Q,  will  be 

P  •  — — ^  •  sm  (p. 
The  vertical  force  applied  in  the  plane  at  B  will  be 


APPLICATIONS.  391 

and  the  horizontal  force  in  this  plane  will  be 

P 
P  •  y-  •  sm  (p  ; 

whence, 

N=-^  V  [^v{l-9)+Q{l-q)  +  P{l-p)coscf^'^+p■'^{l-pY.s\n^^  •  ■  (568) 

N'=  j'Vl^'9  -^  Q.q  +  P.  p.  cos  (p]^  +  P^  .p^  .  sin2  9  ;  .  •  (5G9) 

If  d  and  6'  be  the  angles  -which  the  directions  of  iV  and  jV'  make 
with  that  of  the  resistance   Q,  we  have 

.   ,      F{i-p)     .  .   ^,      i";^     . 

^^"    ^      i\r.  /     '  ^"^  "P '    ^"^         W  '  ^^"  "^^ 

Equations  (5G7),  (568),  and  (569)  are  sufficient  to  determine  the  rela- 
tion between  P  and  Q  to  preserve  the  motion  uniforn*,  or  an  equili- 
brium without  the  aid  of  inertia.  The  values  of  iV^  and  N'  beinf^ 
substituted  in  Equation  (567),  and  that  equation  solved  with  refer- 
ence to  P,  will  give  the  relation  in  question. 

§333. — If  the  power  P  act  in  the  direction  of  the  resistance  Q, 
then  will  cos  9  ==  1,  sin  <p  =  0,  and  Equation  (567)  would,  after 
substituting  the  corresponding  values  of  N  and  N\  transposintr, 
omitting    the    common   factor  d  s^ ,   and   supposing   p  =  p',   become 

PR  =  Qr  +f'p{tv  +   (2  +  P)  +  d^.K±19..r.  .  .   (570) 

And  omitting  the  terms  involving  the  friction  and  stiffness  of 
cordage, 

Q  ~  'R' 

that  is,  the.  power  is  to  the  resistance  as  the  radius  of  the  arbor 
is  to  that  of  the  wheel ;  which  relation  is  exactly  the  same  as 
that   of  the   common   lever. 

FIXED   PULLEY. 

§  334. — The  pulley  is  a  small  wheel  having  a  groove  in  its  cir- 
cumference for   the   reception   of  a  rope,    to   one   end   of  which   the 


392 


ELEMENTS    OF    ANALYTICAL    MECHANICS, 


power  P  is    applied,  and   to  the  other  the  resistance   Q.     The  pulley 
may  turn  either  upon  trunnions  or  about  an  axle,  supported  in  what 


is  called  a  block.  This  is  usually  a  solid  piece  of  wood,  through 
which  is  cut  an  opening  large  enough  to  receive  the  pulley,  and 
allow  it  to  turn  freely  between  its  cheeks.  Sometimes  the  block  is 
a  simple  framework  of  metal.  When  the  block  is  stationary,  the 
pulley  is  said  to  be  fixed.  The  principle  of  this  machine  is  obvi- 
ously the  same   as  that   of  the  wheel  and  axle. 

The  friction  between  the  rope  and  pulley  will  be  sufficient  to 
give  the   latter   motion. 

Making,  in  Equations  (568)  and  (569), 


g  -  q=p 


\l. 


we   have 


N  =\  y/{xo  +  ^  +  P  cos  9)2  +  P2  sin2  9  -'N'  '  -  (571) 


Making  R  =  r,  and  p  =  p',  in  Equation  (567),  and  substituting 
the  above  values  of  iV  and  JV',  we  have,  after  omitting  the  common 
factor  c?Si, 


PR-  QR-f'pV{"'+  §+i'cos(p)2+p2sin2(p-rf^ 


2R 


■R=0.'  (572) 


AT  PLICATIONS.  393 

Solving  this  equation  with  respect  to  P,  we  find  the  value-  of 
the  latter  in  terms  of  the  different  sources  of  resistance.  But  this 
direct  process  would  be  tedious ;  and  it  will  be  sufficient  in  all 
cases  of  practice  to  employ  an  approximate  value  for  P  under  the 
radical,  obtained  by  first  neglecting  the  terms  involving  friction  and 
stiffness  of  cordage. 

Thus,  dividing  by  R  and    transposing,  we  find 

P  =  g  +/'|-  V{^"  -\-  Q  +  i"  <--"^  9)-  +  P^  sin"  9  +  c/,  »  ^  ^^  '^- 

Now  /'  •  —    is    usually  a   small  fraction  ;   an   erroneous   value  as- 

sumed  for  P  under  the  radical,  will  involve  but  a  trifling  error  in 
the  result.  We  may  therefore  write  Q  for  P  in  the  second  mem- 
ber;  and  neglecting  the  weight  of  the  pulley,  which  is  always  in- 
significant  in    comparison   to   Q,  we   have 

P=Q\\  +/-^V^ri  +C0S9)]  +  dr^-^Ji^  '  ■  ■  ^^^^^ 

but 

1  +  cos  9  =  2  cos^  i  9  ; 

whence, 

P=g(l  +2/'-|--cosi^)  +  c/,.^-^  •     •     •     •     (574) 

In  which  9  denotes  the  angle  A  M  B,  which 
is  the  supplement  of  the  angle  A  C  B,  and  de- 
noting this  latter  angle  by  d,  we  have 

cos  i  9  =  sin  ^  ^  , 
whence 

P^Q(l+2f'j^smld)+d,:^~^^-     .    (575) 

If  the  arc  of  the  pulley,  enveloped  by  the  rope,  be  180°,  then 
will 

P=(2(l+2/'.}p  +  J,.^l±^ (576) 


394 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


If  the  friction  and  stiffness  of  cordage  be  so  small  as  to  justify  their 
omission,  then  will 

P=  Q. 

That   is,  the   power   must    be    equal    to  the   resistance,  and    the  only 
office  of  the  cord  or  rope  is  to  change  the  direction  of  the  power. 


MOVABLE    PULLEY. 

§  335. — In  the  fixed  pulley,  the  resultant  action  of  the  power  and 
resistance  is  thrown  upon  the  trunnion  boxes.  If  one  end  of  the 
rope  be  attached  to  a  fixed  hook  A, 
while  the  power  P  is  applied  to  the 
other,  and  the  pulley  is  left  free  to  roll 
along  the  rope,  the  resistance  W  to  be 
overcome  may  be  connected  with  its 
trunnion,  after  the  manner  of  the  figure ; 
the  pulley  is  then  said  to  be  movable, 
and  the  relation  between  the  power  and 
resistance  is  still  given  by  Eq.  (507,) 
in  which  the  principal  resistance  be- 
comes N  -\-  N',  and  the  tension  of  the 
rope  between  the  fixed  point  A,  and  the 
tangential  point  H,  becomes   Q. 

Making  in  Equation  (567),  72  =  r,  p  =  p',  and  IT  =  iV^+  iV^'=  2iV; 
we  have 

K+  IQ 


PR-  QR-f'^W-d, 
dividing  by  i2,  and  transposing 


2R 


2lf 


■  R  =  0 


(577) 


(578) 


Eliminating  Q  by  means  of  Equation  (571),  and  solving  the  resulting 
equation  with  respect  to  P,  the  value  of  the  power  will  be  known 
in  terms  of  the  resistances.  The  process  may  be  much  abridged  by 
limiting  the  solution  to  an  approximation,  which  will  be  found  suffi- 
cient in  practice. 


APPLICATIONS. 


395 


Neglecting  the  weight  of  tlie  pulley,  which  is  always  insignificant 
in  comparison  with  P  or  Q,  and  making  Q  ■=  P,  which  would  be  the 
case  if  we  neglect   friction    and    stiffness  of  cordage,  Equation    (oTl), 


gives 


and  because 


or. 


N=  \W=  \  Q  V-^  (1  +  cos  <p); 

1  +  cos  cp  =  2  cos2  I  (jj  :=  2  sin^  ^  &, 
ir  =  2  (?  .  sin  ^  (3 ; 


Q 


w 


2  sin  ^  d 


which,  in  Equation  (578),  gives 


^='^{d^u^^''i-^-^'' 


K+  I- 


W 


2  sin  ^  d 


2i2 


(579) 


The    quantity    of    work    is   found   by    multiplying    both   members  by 
i2si,  in  which  Sj   is  the   arc  described  at   the  unit's  distance. 

If  the   arc   enveloped   by  the  rope   be   180°,  then  will  ^d  =  90°, 
sin  ^  ^  =  1,  and 


F=  ^^(i+f'-j^)  +^^>- 


2E 


(580) 


If  the   friction   and   stiffness  of   cordage    be   neglected,   then   will. 
Equation  (579), 

TT  =  2  P  sin  I  5, 
and  multiplying  by  P, 

i2  PT  =  P  .  2  P  .  sin  I  ^  ; 
but 

2  B  sin  ^  6  =  A  B; 
W'hence, 

Ji  .  W  =  P  .  AB; 

that  is,  the  power  is  to  the  resistance  as  the 
radius  of  the  ^Jii^/ey  is  to  the  cord  of  the  arc 
enveloped  hij  the  rope. 


596 


ELEMENTS     OF    ANALYTICAL    MECHANICS 


'L 


§  336. — The  Muffie  is  a  collection  of  pulleys  in  two  separate 
blocks  or  frames.  One  of  these  blocks  is  attached  to  a  fixed  point 
yl,  by  which  all  of  its  pulleys  become  fixed^ 
while  the  other  block  is  attached  to  the  resist- 
ance  TF",  and  its  pulleys  thereby  made  mov- 
able. A  rope  is  attached  at  one  end  to  a  hook 
h  at  the  extremity  of  the  fixed  block,  and  is 
passed  around  one  of  the  movable  pulleys, 
then  about  one  of  the  fixed  pulleys,  and  so  on, 
in  order,  till  the  rope  is  made  to  act  upon  each 
pulley  of  the  combination.  The  power  P  is 
applied  to  the  other  end  of  the  rope,  and  the 
pulleys  are  so  proportioned  that  the  parts  of 
the  rope  between  them,  when  stretched,  are 
parallel.  Now,  suppose  the  power  P  to  main- 
tain in  uniform  motion  the  point  of  applica- 
tion of  the  resistance  W\  denote  the  tension 
of  the  rope  between  the  hook  of  the  fixed 
block  and  the  point  where  it  comes  in  con- 
tact with  the  first  movable  pulley  by  t^  \  the 
radius  of  this  pulley  by  i2, ;  that  of  its  eye 
by  r^\  the  co-efficient  of  friction  on  the  axle 
by  /;  the  constant  and  co-efficient  of  the  stiff- 
ness of  cordage  by  K  and  /,  as  before ;  then,  denoting  the  tension  of 
the  rope  between  the  last  point  of  contact  with  the  first  movable, 
and  first  point  of  contact  with  the  first  fixed  pulley,  by  ^j,  the  quan- 
tity of  work  of  the  tension   t^  will.  Equation   (515),  be 


3(o) 


IF 


t.^  El  Sj  =  ti  i?i  Si  +  d^ 


2  Pi 


i^i  5i  +  /'  {t,  +  t,)  r,  s, ; 


in  which 


dividing  by  Sj , 


/'  = 


/ 


VT+r 


t,  P,  =  t,P,  +  d,  .  ^±^  •P,-\-f  {h  +  4)  n.    .     (581) 


APPLICATI0X3.  39T 

Again,  denoting  the  tension  of  that  part  of  the  rope  which  passes 
from  the  first  fixed  to  the  second  movable  pulley  by  4,  the  radius 
of  the  first  fixed  pulley  by  i?.j ,  and  that  of  its  eye  by  r,,  we  shall, 
in  like  manner,  have 

4  R...  =  t,  R,  +  d^  ^—  R.  +  /'  (^.  +  h)  r.      .     (582) 

And  denoting  the  tensions,  in  order,  by  ti  and  t, ,  this  last  being 
equal  to  P,  we  shall  have 

t,R,  =  t,R,  +  d,^^^-R,+f'{h  +  U)r,.     .     (583) 

PR.=  t.R.  +  d,^^^R,+r{f,  +  P)r,.     .     (584) 

so  that  we  finally  arrive  at  the  power  P,  through  the  tensions  which 
are  as  yet  unknown.  The  parts  of  the  rope  being  parallel,  and  the 
resistance  W  being  supported  by  their  tensions,  the  latter  may  ob- 
viously be  regarded  as  equal  in  intensity  to  the  components  of  W; 
hence, 

t,  +  t,  +  t,  +  U=  W;      -     .     '     .     •      (585) 

which,  with  the  preceding,  gives  us  five  equations  for  the  determi- 
nation of  the  four  tensions  and  power  P.  This  would  involve  a 
tedious  process  of  elimination,  which  may  be  avoided  by  contenting 
ourselves  with  an  approximation  which  is  found,  in  practice,  to  be 
sufficiently  accurate. 

If  the   friction   and  stiffness   be   supposed   zero,    for   the   moment, 
Equations  (581)  to  (584)  become 

t.2  i?i  =  /i  -S, , 
/g  R^  =  /.,  R.2 , 

PR,  =  UR,; 
from  which    it   is   apparent,  dividing   out    the   radii    J?,,  R.,  R^.  A:c., 


398  ELEMENTS    OF    ANALYTICAL    MECHANICS. 

that  f.^  =  ti,  ti  =  Li,  fi  =  t^,  F  =  t^;  and  hence,  Equation  (585) 
becomes 

whence, 

the  denominator  4  being  the  whole  number  of  pulleys,  movable  and 
fixed.     Had    there   been   n   pulleys,  then  would 

_    W_ 

'  ~~     n 

With  this  approximate  value  of  ti,  we  resort  to  Equations  (581) 
to  (584),  and  find  the  values  of  t.,,  4,  4,  &;c.  Adding  all  these 
tensions  together,  we  shall  find  their  sum  to  be  greater  than  TF, 
and  hence  we  infer  each  of  them  to  be  too  large.  If  we  now 
suppose  the  true  tensions  to  be  proportional  to  those  just  found, 
and  whose  sum  is  TFj  >  W,  we  may  find  the  true  tension  corre- 
sponding to  any  erroneous  tension,  as  t^ ,  by  the  following  propor- 
tion, viz. : 

W 

or,  which   is  the  same  thing,  multiply  each  of  the    tensions  found  by 

W 

the    constant  ratio  -r—^   the   product  will    be    the    true    tensions,  very 

nearly.  The  value  of  t^  thus  found,  substituted  in  Equation  (584), 
will   give    that   of  P. 

Examj)le. — Let  the  radii  R^,  E.^,  E^  and  E^,  be  respectively 
0,26,  0,39,  0,52,  0,25  feet  ;' the  radii  r,  =  r.,  —  r^  =  u  of -the 
eyes  =  0,06  feet ;  the  diameter  of  the  rope,  which  is  white  and 
dry,  0,79  inches,  of  which  the  constant  and  co-efficient  of  rigidity 
are,  respectively,  Jv  =  1,6097  and  /=  0,0319501  ;  and  suppose  the 
pulley  of  brass,  and  its  axle  of  wrought  iron,  of  which  the  co-efficient 
/  =  0,09,  and   the   resistance    W  a  weight  of  2400  pounds. 

Without   friction    and    stiffiiess  of  cordage, 

2400  '*^- 

t^  ^  rZpi  =  600. 


APPLICATIONS. 


399 


Dividing  Equation  (581)  by  i?i,  it   Lecomes,  since  d^  =  1, 

Substituting  the  value  of  i?i ,  and  the  above  value  of  ^i ,  and  regard- 
ing   in    the   last   term    t.,  as    equal    to  /, ,  which  we   may  do,  because 


of  the    small  co-efficient  -z^  /',   we  fnid 


f       600 

1,6097  -f  0,0319501   x  600 


h=  i 


+ 


2  X  (0,20) 


>  =  028,39. 


+  TT^  X  0'09  X  (600  -f  600) 
0,2b 


Again,  dividing    Equation    (582)    by  i?o ,  and    substituting   this   value 
of  ^2  and   that  of  i?,,  we  find 

lbs. 

f,  -  673,59. 

Dividing  Equation  (583)  by  R^ ,  and  substituting  this  value  of  4 ,  as 
well   as    that   of  R^ ,  there  will    result 


Iha. 

t^  =z  709,82 ; 


whence, 


44     44 


600 
-f-  628,39 
+  673,59 

-f  709,82 


.  =  2611,80; 


and 


W 


2400 


2611,80 
which  will  frivc  for  the    true    values   of 


0,919 


/,  =  0,919  X  600  =  551,400 
i,  =  0,919  X  628.39  =  577,490 
^3  =  0,919  X  673,59  =  619,029 
/,  =  0,919  X  709,82  =  652,324 


2400,243 


400 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


The  above  value  for  ^  =  652,324,  in   Equation  (5S4),  will  give,  after 
dividing  by  i?4,  and  substituting  its  numerical   value, 


P  =  < 


+ 


652,324 

1,6097  +  0,03195  x  652,324 


2  X  0,65 


0,06 


+  ^  X  0,09  X  (652,324  +  P) ; 


and  making  in  the  last  factor  P  =  t^  =  652,324,  we  find 

^  lbs.  lbs.  lbs.  lbs. 

P  =  652,324  +  17,270  +  10,831  =  680,425. 

Thus,  without  friction  or  stiffness  of  cordage,  the  intensity  of  P  would 
be  600  lbs. ;  with  both  of  these  causes  of  resistance,  which  cannot  be 
avoided  in  practice,  it  becomes  680,425  lbs.,  making  a  difference  of 
80,425  lbs,,  or  nearly  one-seventh ;  and  as  the  quantity  of  work  of 
the  power  is  proportional  to  its  intensity,  we  see  that  to  overcome 
friction  and  stiffness  of  rope,  in  the  example  before  us,  the  motor 
must  expend  nearly  a  seventh  more  work  than  if  these  sources  of 
resistance  did  not  exist. 


THE    WEDGE. 

§  337. — The  wedge  is  usually  employed  in  the  operation  of  cut- 
ting, splitting,  or  separating.  It  consists 
of  an  acute  right  triangular  prism  ABC. 
The  acute  dihedral  angle  ACb  \s  called 
the  edge ;  the  opposite  plane  face  A  b 
the  back;  and  the  planes  Ac  and  Cb, 
which  terminate  in  the  edge,  the  faces. 
The  more  common  application  of  the 
wedge  consists  in  driving  it,  by  a  blow 
upon  its  back,  into  any  substance  which 
we  wish  to  split  or  divide  into  parts,  in 
such  manner  that  afler  each  advance  it 
shall  be  supported  against  the  faces  of 
the    opening    till    the    work   is   accomplished. 


APPLICATIONS. 


401 


§  338. — The  blow  by  -which  the  wedge  is  driven  forward  will  be 
supposed  perpendicular  to  its  back,  for  if  it  were  oblique,  it  would 
only  tend  to  impart  a  rotary  motion,  and  give  rise  to  complications 
which  it  would  be  unprofitable  to  consider :  and  to  make  the  case 
conform  still  further  to  practice,  we  Avill  suppose  the  wedge  to  be 
isosceles. 

The  wedge  ACB  being  inserted  in  the  opening  a  A  i,  and  in  con- 
tact with  its  jaws  at  a  and  i,  we  know 
that  the  resistance  of  the  latter  will 
be  perpendicular  to  the  faces  of  the 
wedge.  Through  the  points  a  and  b 
draw  the  lines  aq  and  h j^  normal  to 
the  faces  A  C  and  B  C  \  from  their 
point  of  intersection  0  lay  off  the 
distances  0  q  and  0 2^  equal,  respec- 
tively, to  the  resistances  at  a  and  h. 
Denote  the  first  by  Q,  and  the  second 
by  F.  Completing  the  parallelogram 
0  q  nip,  0  m  will  represent  the  re- 
sultant of  the  resistances  Q  and  F. 
Denote  this  resultant  by  it',  and  the 
angle  ACB  of  the  wedge  by  &.  which, 
in    the    quadrilateral    a  0  h  C,    will    be 

equal  to  the  supplement  of  the  angle  a  0  b  zz^  p  0  q,  the  angle  made 
by  the  directions  of  Q  and  F.  From  the  parallelogram  of  forces, 
we  have, 

U'l  -  pi  j^  q^  j^^p  Q  ^o^j)  0  q  -  F'-  ^  q"  -'IF  Q  dosb; 

or. 


R'  =  ^  7>s  +  Q^  -2F  Qcos 


The  resistance  Q  will  produce  a  friction  on  the  face  -.1  C  equal 
to  fQ,  and  the  resistance  F  will  produce  on  the  face  BC  the  fric- 
tion f  F :  these  act  in  the  directions  of  the'  faces  of  the  wedge. 
Produce  them  till  they  meet  in  C,  and  lay  ofi'  the  distances  C  q'  and 
Cp'  to    represent    their    intensities,    and    complete    the   parallelogram 

26 


402  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

Cq'  0'  'p' ;  G  0'  will  represent  the  resultant  of  the  frictions.  Denote 
this  by  B'\  and  we  have,  from  the  parallelogram  of  forces, 

J?"2  =  p  q^  J^  p  p2  J^  op  p  Qcos6; 
or, 

R"  =  f  -^F^  +  Q'-  +  2  F  Q  cos  6. 

The  wedge  being  isosceles,  the  resistances  F  and  Q  will  be  equal, 
their  directions  being  normal  to  the  faces  will  intersect  on  the  line 
CJ),  which  bisects  the  angle  C  =^  6,  and  their  resultant  will  coin- 
cide with  this  line.  In  like  manner  the  frictions  will  be  equal,  and 
their  resultant  will  coincide  with  the  same  line.  Making  Q  and  F 
equal,  we  have,  from  the  above  equations, 

B'  =     F  ^2(1  —  cos^). 


R"  -fF  ^/'2{\  +  cos  a). 

Bui, 

1  —  cos  ^  =z  2  sin2  1  ^, 

1  +  cos  5  =  2  cos2  ^  ^  . 
whence  we  obtain,  by  substituting  and  reducing, 
R'   =  2  P.  sin  i  ^, 
R"  =  2/.  F.  cos  J-  &  ; 


and  further. 

sm  \&  =  \  ^^  ^„ 

therefore, 

R'-      P      ^^ 

Denote  by  F  the   intensity  of  the  blow  on   the  back  of  the  wedge. 
If  this   blow  be  just    sufficient    to  produce   an   equilibrium    bordering 


APPLICATIONS.  403 

on  motion  forward,  call  it  F' ;  the  friction  will  oppose  it,  and  we 
must  have,  • 

i^'  =  ii;'  +  i2"  =  p.4^  + 2/.p.4^-   •    •    •    (580) 

If,  on  the  contrary,  the  blow  be  just  sufficient  to  prevent  the  wedge 
from  flying  back,  call  it  F" ;  the  friction  will  aid  it,  and  we  must 
have. 

The  Avedge  will  not  move  under  the  action  of  any  force  whose  inten- 
sity is  between  F'  and  F".  Any  force  less  than  F'\  will  allow  it 
to  fly  back  ;  any  force  greater  than  F'^  will  drive  it  forward.  The 
range  through  which  the  force  may  vary  without  producing  motion, 
is  obviously, 

F'-F"^^fP.^ (588) 

which  becomes  greater  and  greater,  in  proportion  as  (7Z)  and  A  C 
become  more  nearly  equal ;  that  is  to  say,  in  proportion  as  the 
wedges  becomes  more  and  more  acute. 

Tlie  ordinary  mode  of  employing  the  wedge  requires  that  it  shall 
retain  of  itself  whatever  position  it  may  be  driven  to.  This  makes 
it  necessary  that  F"  should  be  zero  or  negative,  Eq.  (587),  whence 

A  B        ..  .    ^     C  D  ^AB        ^  ^    ^     C  D 


p —  'Z  f-  P  •  - or   P  ■ <r  'Z,  f  •  P 


i  C  '  AC  A  (J    ^     '  A  C   ' 

or,  omitting   the  common    fictors    and   dividing  both  members  of  the 
equation   and    inequality  by  2  C  D, 

^AB       .  i^^^. 


C  D      ■"  CD 


AB 


but     ^       ^  is   the  tangent  of  the   angle  A  C  D ;  hence  we  conclude, 

that  the  wedge  will  retain  its  place  when  its  semi-angle  does  not 
exceed  that  whose  tangent  is  the  co-efficient  of  friction  between  the 
surface  of  the  wedge  and  the  surface  of  the  opening  which  it  is 
intended  to  cnlarore. 


404 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


Resuming  Eq.  (587),  and  supposing  the  last  term  of  the  second 
member  gi-eater  than  the  first  term,  F''  becomes  negative,  and  will 
represent  the  intensity  of  the  force  necessary  to  withdraw  the  wedge ; 
which  will  obviously  be  the  greatest  possible  when  ^  ^  is  the  least 
possible.  This  explains  why  it  is  that  nails  retain  with  such  perti- 
nacity their  places  when  driven  into  wood,  &c. 


THE   SCREW. 


§  339. — The  Screw,  regarded  as  a  mechanical  power,  is  a  device  by 
which  the  principles  of  the  inclined  plane  are  so  applied  as  to  pro- 
duce considerable  pressures  with  great  steadiness  and  regularity  of 
motion. 

To  form  an  idea  of  the  figure  of  a  screw  and  its  mode  of  action, 
conceive  a  right  cylinder,  a  k,  with  circular  base,  and  a  rectangle,  or 
other  plane  figure,  abc m,  having  one  of  its  sides 
ab  coincident  with  a  surface  element,  while  its 
plane  passes  through  the  axis  of  this  cylinder. 
Next,  suppose  the  plane  of  the  generatrix  to 
rotate  uniformly  about  the  axis,  and  the  gener- 
atrix itself  to  move  also  uniformly  in  the  direc- 
tion of  that  line  ;  and  let  this  twofold  motion 
of  rotation  and  of  translation  be  so  regulated, 
that  in  one  entire  revolution  of  the  plane,  the 
generatrix    shall    jjrogress    in    the    direction    of 

the  axis  over  a  distance  greater  than  the  side  a  b,  which  is  in  the 
surface  of  the  cylinder.  The  generatrix  will  thus  generate  a  pro- 
jecting and  winding  solid  called  a  Jillet,  leaving  between  its  turns 
a  groove  called  the  channel.  Each  point  as  m  in  the  perimeter 
of  the  generatrix,  will  generate  a  curve  called  a  helix,  and  it  is 
obvious,  from  what  has  been  said,  that  every  helix  will  enjoy  this 
property,  viz.  :  any  one  of  its  points  as  m,  being  taken  as  an  origin 
of  reference,  as  well  for  the  curve  itself  as  for  its  projection  on  a 
plane  through  this  point  and  at  right  angles  to  the  axis,  the  distances 
d'm',d"m",  &c.,  of  the  several    points  of  the  helix  from    this  plane, 


APPLICATIOXS. 


405 


are  respectively  proportiuiied  to  the  circular  arcs  md\  md",  &c., 
into  which  the  portions  mm\  mm'\  (Sec,  of  the  helix,  between  the 
origin   and    these    points,    are   projected. 

The  solid  cylinder  about  which  the  fdlet  is  wound,  is  called 
the  newel  of  the  screw;  the  distance  mm'",  between  the  consecu- 
tive turns  of  the  same  helix,  estimated  in  the  direction  of  the  axis, 
is    called    the    helical    interval. 

The  fillet  is  often  generated  by  the  motion  of  a  triangle  with 
one  of  its  sides  coincident  with  ab\  and  as  the  discussion  will  be 
moi'e  general  by  considering  this  mode  of  generation,  we  shall  adopt 
it.  The  surfaces  of  the  fillet,  which  are  generated  by  the  inclined 
faces  of  the  triangle,  are  each  made  up  of  an  infinite  number  of 
helices,  all  of  which  have  the  same  interval,  though  the  helices 
themselves  are  at  diflerent  distances  from  the  axis,  and  have  different 
inclinations   to    that  line. 

The  inclination  of  the  different  helices  to  the  axis  of  the  screw 
increases  from  the  newel  to  the  exterior  surface  of  the  fillet 
the  same  helix  preserving  its 
inclination  unchanged  throughout. 
The  screw  is  received  into  a  hole 
in  a  solid  piece  B  of  metal  or 
wood,  called  a  nut  or  burr.  The 
surface  of  the  hole  through  the 
nut  is  furnished  Avith  a  winding 
fillet  of  the  same  shape  and  size 
as  the  channel  of  the  screw,  so 
that  the  surfaces  of  the  screw  and 
nut  are  brought  into  accurate  con- 
tact. 

From  this  arrangement  it  is 
obvious  that  when  the  nut  is  sta- 
tionary, and  a  rotary  motion  is 
communicated  to  the  screw,  the 
latter  will    move    in    the   direction 

of    its    axis ;     also,    when    the    screw    is    stationary    and    the   nut    is 
turned,   the   nut    must   also   move   in  the   direction   of  the   axis.      In 


406 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


the  first  case,  one  entire  revolution  of  the  screw  will  carry  it  lon- 
gitudinally through  a  distance  equal  to  the  helical  interval,  and  any 
fractional  portion  of  an  entire  revolution  will  carry  it  through  a  pro- 
portional distance;  the  same  of  the  nut,  when  the  latter  is  mova- 
ble and  the  screw  stationary.  The  resistance  Q  is  applied  either  to 
the  head  of  the  screw,  or  to  the  nut,  depending  upon  which  is  the 
movable  element ;  in  either  case  it  acts  in  the  direction  i)  C  of 
the  axis.  The  power  F  is  applied  at  the  extremity  of  a  bar  GH 
connected  with  the  screw  or  nut,  and  acts  in  a  plane  at  right 
angles  to   the   axis   of  the   screw. 

From    the    description   of  the   screw  and   its  mode   of  generation, 
we   may  find   the    equation    of   its   fillet   or   helicoidal    surface.      Tor 
this  purpose,  take  the  axis  z  to  coincide  with  the   axis  of  the  newel, 
and   the   initial   position  of  the   generatrix  in   the   plane  yz.     Make 
6-   =  any    definite    portion    C  C 

of  an  assumed  helix  ; 
9  =  the    angle    YA  t,    through 
which    the    rotating    plane 
has  turned  during  the  gene- 
ration of  s ; 
r  =  the    distance    C  D    of    this 

helix  from    the  axis  z; 
a  =  the   angle  which  this   helix 
makes  with  the  plane  xy; 
§  =z  the  angle   CBD  which  the 
generatrix  of  the  helicoidal 
surface     makes     with     the 
axis  z ; 
y  =z  the  co-ordinate  AB  of  the 
point  in  which   the  genera- 
trix, in  its  initial   position,  intersects   the   axis  z. 
Then,  for  any  point   as   C  of  the  generatrix  in  its  initial  position, 

we   have 

z  —  AI>=AB  +  BD  =  'y  +  r.  cotan  §, 

and   for   any  subsequent  position,  as   C  B', 

z  =  y  +  r  .  cotan  €  +  r  .  o  .  tan  a,  .     •     •     •     (589) 


APPLICATIONS.  407 

■which  is  the  equation  souglit,  and  in  wlilch  a  and  r  are  constant 
for   the   same   helix,  and  variable  from  one  helix   to  another. 

The   power   P   acts  in   a   direction    perpendicular    to   the   axis  of 
the  newel.     Denote   by  I  its   lever  arm  ;  its  virtual   moment  will  be 

Pld^. 

The  resistance  Q  acts  in  the  direction  of  the  axis  of  the  newel ; 
its   virtual   moment  will   be 

Qdz. 

The  friction  acts  in  the  direction  of  the  helicoidal  surface  and  paral- 
lel to  the  helices.  Conceive  it  to  be  concentrated  upon  a  mean 
helix,  of  which  the  distance  from  the  newel  axis  is  r,  and  length  s : 
denote  the  normal  pressure  by  TV,  and  co-efficient  of  friction  by  / 
The  virtual  moment  of  friction  will  be  ■ 

f.lY.ds; 
and  Equation  (508), 

Pld(p  -  Qdz -f.N .ds  =  0 (590) 

But   the  displacement    must   satisfy  Equation    (589),  or,  as  in  §  213,    * 
the  condition, 

L  =  z  —  r  .  Cf  .  tan  a.  —  r  .  cotan  ?  —  y  =  0 ;       .     (591) 

and  also, 

r  =  constant (592) 

Differentiating,  we  have, 

dz  —  cotan  §  .  d  r  —  r  tan  a  d  (p  =  0, 

dr  =  0. 

Multiplying  the  first  by  X,  the  second  by  X',  adding  to  Equation 
(590),  and  eliminating  d  s  by  the  relation 

d  s  =  r  .  dcp  .  cos  a  -f  (/  z  .  sin  a,    .     .     .     .     (593) 
we  find, 

i^  i  —/■  -i^-cosa  .r  —  \Uxia.r)df  +  {\  -  Q  -  f.A^'.sina)  d  z  +  (\'—  Xcotaa^)d  r  ==  0; 


408  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

and,  from  the  principle  of  indeterminate  co-efficients, 

p  ^  _  / .  iV" .  cos  a  .  r  —  X  .  tan  a  .  r  =  0 ;     .     .     (594) 

Q  -^f:^  .s'ma  —  \  =  0; (595) 

X'  —  X  cotan  g  =  0 (595)' 

The    variables   d  z,  d  r,  and  d  9,  are    rectangular  ;    whence.    Equation 
(331), 

M=.^  (^)V  (i^)\  (^)'-=  X  VI  +  .-^  «  +  cou,«. 

Substituting  this  in  Equations  (594)  and  (595),  and  eliminating  X, 
there  will  result 

r      tan  a  +  /.  cos  a  .  -Ji  +  tan^  a  +  cotan^  g 

J>  =z    Q  .  —  . *  -  — -      [i)vb) 

I  1  —  / .  sin  a  .  -/l  +  tan2  „  +  cotan'^  S 

Substituting    the    value   of  X   from   Equation    (595),  in    Equation 

(595)',  we  find, 

cotan  §  f-n<y\ 

X' =  Q ^.==-;    .     (o9/) 

1  —  / .  sin  a  y  1  +  tan2  a  +  cotan^  5 

in  which  X'  is,  §  217,  the  value    of  the    force   acting    in    the  direction 
of  r. 

R340._If  the  fillet   be   rectangular,  §  =  90°,  cotan  g  =  0,  and 


p=  Q  .L.  tan  g  +  /.  cos  g  .  yTj^tan^  a  _      ^     ^^^^^ 
I  1  —  /  .  sin  g  .  -/l  +  tan^  a 

and 

X'  ==  0. 

§341. — If  we  neglect  the  friction,  /  =  0  ;  and 
PI  =z  Q  .  r  .  tan  g, 

multiplying  both  members  by  2  ■jt', 

P  .  2  •TT  ^  =  §  .  2  *  r  .  tan  g (599) 

That   is,   the  power  is  to  the  resistance   as    Uie  helical   interval  is    to 
the  circumference  described  by  the  end  of  the  lever  arm  of  the  power. 


APPLICATIONS. 


409 


ruMPS. 


g  34o_ — Any  macliine  used  fur  raising  liquids  from  one  level 
to  a  higher,  in  which  the  agency  of  atmospheric  pressure  is  employed, 
is  called  a  Pump.  There  are  various  kinds  of  pumps ;  the  more 
common  are  the  sucklnff,  forcing,  and  Ufllng  pumps. 

I  343. — The  Sucking-Pum]}  consists  of  a  cylindrical  body  or  barrel 
B,  from  the  lower  end  of  which  a  tube  B,  called  the  sucking-pipe, 
descends  into  the  water  contained  in  a  reservoir  or  well.  In  the 
interior  of  the  barrel  is  a  movable  piston  C,  surrounded  with  leather 
to  make  it  water-tight,  yet  ca- 
pable of  moving  up  and  down 
freely.  The  piston  is  perforated 
in  the  dii'cction  of  the  bore  of 
the  barrel,  and  the  orifice  is 
covered  by  a  valve  I^  called 
the  piston-valve,  which  opens  up- 
ward ;  a  similar  valve  £,  called 
the  sleeping-valve,  at  the  bottom 
of  the  barrel,  covers  the  upper 
end  of  the  sucking-pipe.  Above 
the  highest  point  ever  occupied 
by  the  piston,  a  discharge-pipe 
P  is  inserted  into  the  barrel  ; 
the  piston  is  worked  by  means 
of  a  lever  //,  or  other  contriv- 
ance, attached   to  the  piston-rod 

G.  The  distance  A  A',  between  the  highest  and  lowest  points  of  the 
piston,  is  called  the  plaij.  To  explain  the  action  of  this  pump,  let 
the  piston  be  at  its  lowest  point  A,  the  valves  U  and  F  closed  by 
their  own  weight,  and  the  air  within  the  pimip  of  the  same  density 
and  elastic  force  as  that  on  the  exterior.  The  water  of  the  reservoir 
will  stand  at  the  same  level  L  L  both  within  and  without  the 
sucking-pipe.  Now  suppose  the  piston  raised  to  its  highest  point  A', 
the    air    contained    in    the    barrel    and    sucking-pipe    will    tend    by   its 


410         ELEMENTS     OF    ANALYTICAL    MECHANICS. 

elastic  force  to  occupy  the  space  Avhich  the  piston  leaves  void,  the 
valve  E  will,  therefore,  be  forced  open,  and  air  will  pass  from  the 
pipe  to  the  barrel,  its  elasticity  diminishing  in  proportion  as  it  fills 
a  larger  space.  It  will,  therefore,  exert  a  less  pressure  on  the 
water  below  it  in  the  sucking-pipe  than  the  exterior  air  does  on  that 
in  the  reservoir,  and  the  excess  of  pressure  on  the  part  of  the 
exterior  air,  will  force  the  water  up  the  pipe  till  the  weight  of  the 
suspended  column,  increased  by  the  elastic  force  of  the  internal  air, 
becomes  equal  to  the  pressure  of  the  exterior  air.  When  this  takes 
place,  the  valve  E  will  close  of  its  own  weight ;  and  if  the  piston 
be  depressed,  the  air  contained  between  it  and  this  valve,  having 
its  density  augmented  as  the  piston  is  lowered,  will  at  length  have 
its  elasticity  greater  than  that  of  the  exterior  air;  this  excess  of 
elasticity  will  force  open  the  valve  F,  and  air  enough  will  escape 
to  reduce  what  is  left  to  the  same  density  as  that  of  the  exterior 
air.  The  valve  F  will  then  fall  of  its  own  weight ;  and  if  the 
piston  be  again  elevated,  the  water  will  rise  still  higher,  for  the 
same  reason  as  before.  This  operation  of  raising  and  depressing 
the  piston  being  repeated  a  few  times,  the  water  will  at  length  cntei 
the  barrel,  through  the  valve  F,  and  be  delivered  from  the  dis- 
charo-e-pipe  P.  The  valves  E  and  F,  closing  after  the  water  has 
passed  them,  the  latter  is  prevented  from  returning,  and  a  cylinder 
of  water  equal  to  that  through  which  the  piston  is  raised,  will,  at 
each  upward  motion,  be  forced  out,  provided  the  discharge-pipe  is 
laro-e  enough.  As  the  ascent  of  the  water  to  the  piston  is  pro- 
duced by  the  difference  of  pressure  of  the  internal  and  external  air, 
it  is  plain  that  the  lowest  point  to  which  the  piston  may  reach, 
should  never  have  a  greater  altitude  above  the  water  in  the  reser- 
voir than  that  of  the  column  of  this  fluid  which  the  atmospheric 
pressure    may    support,    in    vacuo,    at   the    place. 

g  344. — It  will  readily  appear  that  the  rise  of  water,  during 
each  ascent  of  the  piston  after  the  first,  depends  upon  the  expulsion 
of  air  through  the  piston-valve  in  its  previous  descent.  But  air  can 
only  issue  through  this  valve  when  the  air  below  it  has  a  greater 
density    and    therefore   greater    elasticity    than   the    external   air ;    and 


APPLICATIONS. 


411 


if  the  piston  may  not  descend  low  enough,  for  want  of  sufficient 
play,  to  produce  this  degree  of  compression,  the  water  must  cease 
to  rise,  and  the  working  of  the  piston  can  have  no  other  effect  than 
alternately  to  compress  and  dilate  the  same 
air  between  it  and  the  surface  of  th(f  water. 
To  ascertain,  therefore,  the  relation  which  the 
play  of  the  piston  should  bear  to  the  other 
dimensions,  in  order  to  make  the  pump  effec- 
tive, suppose  the  water  to  have  reached  a  sta- 
tionary level  X,  at  some  one  ascent  of  the 
piston  to  its  highest  point  A',  and  that,  in  its 
subsequent  descent,  the  piston-valve  will  not 
open,  but  the  air  below  it  will  be  compressed 
only  to  the  same  density  with  the  external  air 
when  the  piston  reaches  its  lowest  point  A. 
The  piston  may  be  worked  up  and  down  in- 
definitely, within  these  limits  for  the  play, 
■without   moving   the  water.      Denote    the  play 

of  the  piston  by  a;  the  greatest  height  to  which  the  piston  may  be 
raised  above  the  level  of  the  water  in  the  reservoir,  by  i,  which  may 
also  be  regarded  as  the  altitude  of  the  discharge  pipe ;  the  elevation 
of  the  point  X,  at  which  the  water  stops,  above  the  water  in  the 
reservoir,  by  x ;  the  cross-section  of  the  interior  of  the  barrel  by  B. 
The  volume  of  the  air  between  the  level  X  and  A  will  be 


B  X  {b  —  X  —  a); 

the  volume  of  this  same  air,  when  the  piston  is  raised  to  A',  pro- 
vided the  water   does  not   move,  will    be 

B{b  -  x). 

Eepresent  by  h  the  greatest  height  to  wnich  water  may  be  supported 
in  vacuo  at  the  place.  The  weight  of  the  column  of  water  which 
the  elastic  force  of  the  air,  when  occupying  the  space  between  the 
limits  X  and  A,  will  sujiport  in  a  tube,  with  a  bore  equal  to  that 
of  the  barrel    is  measured   by 

Bk.g.  Z>; 


412  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

in  which  D  is  the  density  of  the  water,  and  g  the  force  of  gravity. 
The  weight  of  the  column  which  the  elastic  force  of  this  same  air 
will    support,  when    expanded   between    the   limits  X  and  A\  will  be 

Bh'  .g  .D; 

in  which  h'  denotes  the  height  of  this  new  column.  But,  from  Ma- 
riotte's    law,  we  have 

B{h  -  X  -  a)  :  B{b  -  x)  :  :  B  W  g  D  :  B  h  g  D  ; 

whence, 

h  —  X  —  a 


h'  =  h 


b  —  X 


But  there  is  an  equilibrium  between  the  pressure  of  the  external 
air  and  that  of  the  rarefied  air  between  the  limits  X  and  A\  when 
the  latter  is  increased  by  the  weight  of  the  column  of  water  whose 
altitude  is  x.     Whence,  omitting  the  common  factors  B,  B  and  y, 

X  ■\-  h    =  X  -\-  h-  — ; =  k : 

0  —  X 

or,  clearing    the  fraction  and    solving    the    equation  in   reference  to  x, 
we  find 


X  =  ib  ±  ^  ^^2  -  4:  ah. (600) 

When   X   has    a   real    value,  the    water    will    cease    to    rise,  but   x 
will   be    real    as    long   as    b^   is   greater    than    4  a  h.      If,  on    the    con- 
trary, 4a^  is  greater  than  b^,  the  value  of  x  will  be  imaginary,  and 
the  water  cannot  cease  to  rise,  and  the  pump  will  always  be  effective 
hen   its    dimensions    satisfy  this   condition,  viz,  : — 

4a  h  y  b-, 
or, 

">4l' 

that  is  to  say,  the  lilay  of  the  j^iston  must  be  greater  than  the  square 
of  the  altitude  of  the  iqyper  limit  of  the  j^/ay  of  the  2)iston  above 
the  surface  of  the  ivater  in  the  reservoir,  divided  by  four  times  the 
height    to   ivhich    tlie  atmospheric  pressure  at  the  place,   where   the  pump 


APPLICATIONS. 


413 


is  zised,  will  support  water  in  vacuo.  This  last  heiglit  is  easily  found 
by  means  of  the  baronicter.  We  have  but  to  notice  the  altitude 
of  the  barometer  at  the  place,  and  multiply  its  column,  reduced  to 
feet,  by  13^,  this  being  the  specific  gravity  of  mercury  referred  to 
water  as  a  standard,  and  the  product  will  give  the  value  of  h  in 
feet. 

JExample. — Required  the  least  jilay  of  the  piston  in  a  suckino-. 
pump  intended  to  raise  water  through  a  height  of  13  feet,  at  a 
l^lace  where    the    barometer   stands   at   28    inches. 


Here 


b  =  13,     and     b^~  z=  169. 


28 
12  ■ 

= 

2,333 

feet. 

h  = 

2,333 

X 

13,5 

=  31 

,5  feet. 

=  a  > 

62 

4h  ~ 

169 

ft. 
1,341  +  ; 

4 

X  31 

o 

Barometer, 


Play 


that   is,  the   play  of  the   piston   must   be  greater   than   one   and   one 
third  of  a   foot. 

§345. — The  quantity  of  work  performed  by 
the  motor  during  the  delivery  of'  water  through 
the  discharge-pipe,  is  easily  computed.  Sup- 
pose the  piston  to  have  any  position,  as  J/, 
and  to  be  moving  upward,  the  water  being 
at  the  level  L  L  in  the  reservoir,  and  at  P 
in  the  pump.  The  pressure  upon  the  upper 
surfece  of  the  piston  will  be  equal  to  the 
entire  atmospheric  pressure  denoted  by  A^ 
increased  by  the  weight  of  the  column  of 
water  MP',  whose  height  is  c',  and  whose 
base  is  the  area  B  of  the  piston  ;  that  is,  the 
pressure   upon    the   top  of  the   piston  will  be 

A  +  Be  cjD, 

in  which  (j  and  i>  are  the  force  of  gravity  and  density  of  the  water, 
respectively.      Again,    the    pressure    upon    the    under    surface    of    the 


1 


M 


414  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

piston  is  equal  to  the  atmospheric  pressure  A,  transmitted  through 
the  water  in  the  reservoir  and  up  the  suspended  column,  diminished 
by  the  weight  of  the  column  of  water  JVM  below  the  piston,  and 
of  which  the  base  is  B  and  altitude  c ;  that  is,  the  pressure  from 
below  will   be 

A  —  BcgD, 

and   the    difference  of  these   pressures  will    be 

A-\-  Be'  gD  -  {A  -  Beg  D)  z:^  BgD{c  +  c') ; 
but,  employing  the   notation    of  the   sucking-pump  just   described, 

c  ^  c'  =  h; 
whence,  the  foregoing   expression   becomes 

Bb.g.D', 

which  is  obviously  the  weight  of  a  column  of  the  fluid  whose  base 
is  the  area  of  the  piston  and  altitude  the  height  of  the  discharge-pipe 
above  the  level  of  the  water  in  the  reservoir.  And  adding  to  this 
the  effort  necessary  to  overcome  the  friction  of  the  parts  of  the  pump 
when  in  motion,  denoted  by  9,  we  shall  have  the  resistance  which  the 
force  F,  applied  to  the  piston-rod,  must  overcome  to  produce  any 
useful  effect ;  that  is, 

F  =  BbgD  +  9. 

Denote  the  play  of  the  piston  by  ^^,  and  the  number  of  its  double 
strokes,  from  the  beginning  of  the  flow  through  the  discharge-pipe 
till  any  quantity  Q  is  delivered,  by  n ;  the  quantity  of  work  will,  by 
omitting* the  effort  necessary  to  depress  the  piston,  be 

Fnp  =  11 2^  \_B  h  .  g  D  -\-  ^'\; 

or  estimating  the  volume  in  cubic  feet,  in  which  case  p  and  h  must 
be  expressed  in  linear  feet  and  B  in  square  feet,  and  substituting  for 
g  D  its  value  62,5  pounds,  we  finally  have  for  the  quantity  of  work 
necessary  to  deliver  a  number  of  cubic  feet  of  water   Q  =  B  n}^, 

Fnp  =  np  [G2,.5  .Bb  +  cp];      .     .     .     .     (601) 

in   which   9    must   be   expressed    in    pounds,  and    may  be  determined 


APPLICATIONS. 


415 


either  by   experiment  in  each  particular   puinp,  or   computed    by  the 
rules  already  given. 

It  is  apparent  that  the  action  of  the  sucking-pump  must  be  very 

%  irregular,   and  that  it  is  only  during  the  ascent  of  the  piston  that  it 

produces  any  useful  effect;  during  the  descent  of  the  piston,  the  force 

is  scarcely   exerted  at  all,   not   more   than   is   necessary  to   overcome 

the  friction. 


§  346. — The  Lifting -Pxivrp  does  not  differ  much  from   the  sucking- 
pump  just  described,  except  that  the  barrel   and  sleeping-valve  E  are 
placed  at  the  bottom  of  the  pipe,  and  some  distance  below  the  sur- 
face of  the  water  L  Lm  the  reservoir ;  the 
piston    may    or    may    not    be    below     this 
same  surface  when  at  the  lowest  point  of 
its  play.      The   piston    and    sleeping-valves 
open  upward.     Supposing  the  piston  at  its 
lowest  point,  it   will,  when   raised,  lift  the 
column    of   water    above    it,   and    the  pres- 
sure of  the   external   air,  together  with  the 
head   of   fluid   in    the    reservoir    above    the 
level   of  the  sleeping-valve,   will   force   the 
latter  open  ;   the   water  will    flow    into    the  '  ^ 

barrel  and  follow    the    piston.       When    the  ^ 

piston  reaches   the  upper  limit  of  its  play,  1 

the    slceping-valvo    will    close    and    prevent 
the   return   of   the   water    above   it.      The 

piston  being  dej^ressed,  its  valves  F  will  open  and  the  water  will 
flow  through  them  till  the  piston  reaches  its  lowest  point.  The 
same  operation  being  repeated  a  few  times,  a  column  of  water  will 
be  lifted  to  the  mouth  of  the  discharge-pipe  P,  after  wliirh  every 
elevation  of  the  piston  will  deliver  a  volume  of  the  fluid  equal  to 
that  of  a  cylinder  whose  base  is  the  area  of  the  piston  and  whose 
altitude  is  equal  to  its  play. 

As  the  water  on  the  same  level  within  and  without  the  pump 
will  be  in  equilibrio,  it  is  plain  that  the  resistance  to  be'  overcome 
by  the  power  will  be  the  friction  of  the  rubbing  surfaces  of  the  pump, 


416 


ELEMENTS     OF     ANALYTICAL    MECHANICS. 


augmented  by  the  weight  of  a  column  of  fluid  whose  base  is  the  area 
of  the  piston,  and  altitude  the  difterenee  of  level  between  the  surface 
of  the  water  in  the  reservoir  and  the  discharge-pipe.  Hence  the 
quantity  of  work  is  estimated  by  the  same  rule,  Equation  (601).  If 
we  omit  for  a  moment  the  consideration  of  friction,  and  take  but  a 
single  elevation  of  the  piston  after  the  water  has  reached  the  dis- 
charge-pipe, n  will  equal  one,  9  will  be  zero,  and  that  equation  re- 
duces to 

F]}  =  G2,5  Bj^   X  b; 

but  62,5  X  Bp  is  the  quantity  of  fluid  discharged  at  each  double 
stroke  of  the  piston,  and  b  being  the  elevation  of  the  discharge-pipe 
above  the  water  in  the  reservoir,  we  see  that  the  work  will  be  the 
same  as  though  that  amount  of  fluid  had  actually  been  lifted  through 
this  vertical  height,  which,  indeed,  is  the  useful  effect  of  the  pump 
for  every   double  stroke. 

§  347. — The  Forcing-Pump 
is  a  further  modification  of 
the  simple  sucking-pump.  The 
barrel  B  and  sleeping-valve 
F  are  placed  upon  the  top 
of  the  sucking-pipe  M.  The 
piston  F  is  without  per- 
foration and  valve,  and  the 
water,  after  being  forced  into 
the  barrel  by  the  atmospheric 
pressure  without,  as  in  the  suck- 
ing-pump, is  driven  by  the  de- 
pression of  the  piston  through 
a  lateral  pipe  H  into  an  air- 
vessel  iV,  at  the  bottom  of 
which  is  a  second  sleeping- valve 
E',  .opening,  like  the  first,  up- 
ward. Through  the  top  of  the 
air-vessel  a  discharge-pipe  K 
passes,   air-tight,   nearly   to    the 


APPLICATIONS.  417 

bottom.  The  Avater,  when  forced  into  the  iiir-vessel  by  the  de- 
scent of  the  piston,  rises  above  the  lower  end  of  this  pipe. 
confines  and  compresses  tlie  air,  which,  reacting  by  its  elas- 
ticity, forces  the  water  up  the  pipe,  while  the  valve  E'  is  closed  by 
its  own  weight  and  the  pressure  from  above,  as  soon  as  the  piston 
reaches  the  lower  limit  of  its  play.  A  few  strokes  of  the  piston  will, 
in  general,  be  sufficient  to  raise  water  in  the  pipe  K  to  any  desired 
height.,  the  only  limit  being  that  determined  by  the  power  at  com- 
mand and  the  strength  of  the  pump. 

§348. — During  the  ascent  of  the  piston,  the  valve  E'  is  closed 
and  E  is  open  ;  the  pressure  upon  the  upper  surface  of  the  piston 
is  that  exerted  by  the  entire  atmosphere ;  the  pressure  upon  the 
lower  surface  is  that  of  the  entire  atmosphere  transmitted  from  the 
surface  of  the  reservoir  through  the  fluid  up  the  jjump,  diminished 
by  the  weight  of  the  column  of  water  whose  base  is  the  area  of 
the  piston  and  altitude  the  height  of  the  piston  above  the  surface 
of  the  water  in  the  reservoir  ;  hence,  the  resistance  to  be  overcome 
by  the  power  will  be  the  difierence  of  these  pressures,  which  is 
obviously  the  weight  of  this  column  of  water.  Denote  the  area 
of  the  piston  by  B^  its  height  above  the  water  of  the  reservoir  at 
one  instant  by  y,  and  the  weight  of  a  unit  of  volume  of  the  fluid 
by  w,  then  will  the  resistance  to  be  overcome  at  this  point  of  the 
ascent  be 

lo.B.y; 

and   the    elementary  quantity  of  work   will   be 

IV  .  B  .ydy; 

and   the  whole  work    during    the   ascent  will    be 

7?  /""'  ^  p  y'  +  y>  t  >       N 

^^ * -^Jy,  y^y  =  ^-^  — ^  (y  -  y^\ 

in  which  y'  and  y^  are  the  distances  of  the  upper  and  lower  limits 
of  the    play  of  the   piston  from    the  water   in    the   reservoir. 

But  B .  [y'  —  y,)  is  the  volume  of  the  barrel  within  the  limits 
of  the  play  of  the  piston,  and  i  [y'  +  y^)  is  the  height  of  its  centre 
of  gravity  above    the   level  of  the  fluid    in    the   reservoir. 


418     ELEMENTS  OF  ANALYTICAL  MECHANICS. 

y'  -\-  y 
Denoting  the    play  by  p,  and   making    — - — '-  =  2',  we   have  for 

the   quantity  of  work    during  the   ascent, 

w.B.p.z'. 

During  the  descent  of  the  piston,  the  valve  E  is  closed,  and  E' 
open,  and  as  the  columns  of  the  fluid  in  the  barrel  and  discharge- 
pipe,  below  the  horizontal  plane  of  the  lower  surface  of  the  piston, 
will  maintain  each  other  in  equilibrio,  the  resistance  to  be  '  over- 
come by  the  power  will  be  the  weight  of  a  column  of  fluid  whose 
base  is  the  area  of  the  piston  and  altitude  the  difference  of  level 
between  the  piston  and  point  of  delivery  P ;  and  denoting  by  z, 
the  distance  of  the  central  point  of  the  play  below  the  point  P, 
we   shall    find,  by  exactly  the    same    process, 

w  B  p  z^  , 

for  the  quantity  of  work  of  the  motor  during  the  descent  of  the 
piston ;  and  hence  the  quantity  of  work  during  an  entire  double 
stroke  will  be    the   sum  of  these,  or 

xoBp{z'  +  z)i. 

But  z'  -f  Zi  is  the  height  of  the  point  of  delivery  P  above  the 
surface  of  the  water  in  the  reservoir ;  denoting  this,  as  before,  by 
5,  we  have 

wBpb;  , 

and  calling  the  number  of  double  strokes  n,  and  the  whole  quantity 
of  work   Q,  we   finally    have 

Q  =  nwBpb. (602) 

If  we    make    z,  =  z',    or   5=2  z^ ,  which  will  give    z^  —  —5    the 

quantity  of  work  during  the  ascent  will  be  equal  to  that  during 
the  descent,  and  thus,  in  the  forcing-pump,  the  work  may  be  equalized 
and  the  motion  made  in  some  degree  regular.  In  the  lifting  and 
sucking-pumps  the  motor  has,  during  the  ascent  of  the  piston,  to 
ovei'come  the  weight  of  the  entire  column  whose  base  is  equal  to 
the  area  of  the   piston   and   altitude   the   difference  of  level   between 


APPLICATIONS. 


410 


the  water  in  the  reservoir  and  point  of  delivery,  and  being  wholly 
relieved  during  the  descent,  when  the  load  is  thruwu  upon  the 
sleeping-valve  and  its  box,  the  work  becomes  variable,  and  the 
motion   irregular. 

THE    SIPHON. 


g  349. — The  Sij^koti  is  a  bent  tube  of  unequal  branches,  open  at 
both  ends,  and  is  used  to  convey  a  liquid 
from  a  higher  to  a  lower  level,  over  an  in- 
termediate point  higher  than  either.  Its 
parallel  branches  being  in  a  vertical  plane 
and  plunged  into  two  liquids  whose  upper 
surfaces  are  at  L  M  and  L'  M\  the  fluid 
will  stand  at  the  same  level  both  within 
and  without  each  branch  of  the    tube   when 

a   vent   or    small   opening    is    made   at    0.  L i 

If    the   air   be    withdrawn    from    the    siphon 

through  this  vent,  the  water  will  rise  in  the 

branches   by  the   atmospheric    pressure    without,    and    when    the   two 

columns  unite  and   the  vent  is  closed,   the  liquid  will  flow   from   the 

reservoir  A  to  A',  as  long  as  the  level  L'  M'  is  below  L  J/,  and  the 

end  of  the  shorter  branch  of  the  siphon    is  below  the    surface    of  the 

liquid   in   the  resei'voir   A. 

The  atmospheric  pressures  upon  the  surfaces  L  M  and  L'  M', 
tend  to  force  the  liquid  up  the  two  branches  of  the  tube.  When 
the  siphon  is  filled  with  the  liquid,  each  of  these  pressures  is  coun- 
teracted in  part  by  the  weight  of  the  fluid  column  in  the  branch 
of  the  siphon  that  dips  into  the  fluid  upon  which  the  pressure  is 
exerted.  The  atmospheric  pressures  are  very  nearly  the  same  for  a 
diftcrence  of  level  of  several  feet,  by  reason  of  the  slight  density 
of  air.  The  weights  of  the  suspended  columns  of  water  will,  for  the 
same  diflorence  of  level,  diff'er  considerably,  in  consequence  of  the 
greater  density  of  the  liquid.  The  atmospheric  pressure  opposed 
to  the  weight  of  the  longer  column  will  therefore  be  more  counter- 
acted  than    that   opposed    to   the   weight  of   the   shorter,   thus  leaving 


420  ELEMENTS     OF    ANALYTICAL    MECHANICS. 

an  excess  of  pressure  at  the  end  of  the  shorter  branch,  which  will 
produce  the  motion.  Thus,  denote  by  A  the  intensity  of  the  at- 
mospheric pressure  upon  a  surface  a  equal  to  that  of  a  cross-section 
of  the  tube ;  by  h  the  difference  of  level  between  the  surface  L  M 
and  the  bend  0;  hy  h'  the  difference  of  level  between  the  same 
point  0  and  the  level  L'  M' ;  by  D  the  density  of  the  liquid; 
and  by  ff  the  force  of  gravity :  then  will  the  pressure,  which  tends 
to   force   the   fluid  up   the   branch   which  dips   below   L  M,  be 

A  —  ah  Dff', 
and    that   which   tends    to   force   the   fluid   up   the  branch   immersed 
in   the   other   reservoir,  be 

A  —  ah'  D  g; 

and  subtracting  the  first  from  the  second,  we  find 

aDg  [h'  -  h), 

for  the  intensity  of  the  force  which  urges  the  fluid  within  the 
siphon,  from  the  upper  to   the   lower  reservoir. 

Denote  by  I  the  length  of  the  siphon  from  one  level  to  the 
other.  This  will  be  the  distance  over  which  the  above  force  will 
be    instantly    transmitted,    and    the    quantity    of   its    work   will    be 

measured   by 

aDg{h'  -  h)l. 

The  mass  moved  will  be  the  fluid  in  the  siphon  which  is  measured 
by  alD\  and  if  we  denote  the  velocity  by  F,  we  shall  have,  for  the 
living  force  of  the  moving  mass, 

alD.  F2; 

whence, 

,          ..,      aDlV^ 
aBgili'  -h)l=  2— 5 

and, 

V  =  V2^(A'  -A); 

from  which  it  appears,  that  the  velocity  tvith  lohich  the  liquid  loill 
fiow  throvgh  the  siphon,  is  equal  to  the  square  root  of  twice  the  force 
of  gravity,  into    the   difference   of  level   of   the  fluid  in   the   two   reser- 


APPLICATIONS.  421 

voirs.  "When  the  fluid  in  the  reservoirs  conies  to  the  same  level, 
the    flow    will    cease,  since,  in  that  case,  h'  —  h  =  0. 

§350. — The    siphon    may    be    employed    to   great    advantage     to 
drain  canals,  ponds,  marshes,  and  the  like.     For  this  purpose,  it  may 
be  made  flexible  by  constructing  it 
of    leather,    well      saturated     with 
grease,  like  the   common   hose,  and  0_^ 

furnished  with  internal  hoops  to 
prevent  its  collapsing  by  the  pres- 
sure   of   the    external    air.      It    is 

throwii  into  the  water  to  be  drained,  "^ 

and  filled ;  when,  the  ends  being 
plugged  up,  it   is   j^laced  across  the 

ridge  or  bank  over  which  the  water  is  to  be  conveyed ;  the  plugs 
are  then  removed,  the  flow  will  take  place,  and  thus  the  atmos- 
phere will  be  made  literally  to  press  the  water  from  one  basin  to 
another,  over   an    intermediate  ridge. 

It  is  obvious  that  the  diflerence  of  level  between  the  bottom  of 
the  basin  to  be  drained  and  the  highest  point  0,  over  which  the 
water  is  to  be  conveyed,  should  never  exceed  the  height  to  which 
water  may  be  supported  in  vacuo  by  the  atmospheric  pressure  at 
the   place. 

THE    AIR-PUMP. 

§351. — Air  expands  and  tends  to  diffuse  itself  in  all  directions 
when  the  surrounding  pressure  is  lessened.  By  means  of  this  pro- 
perty, it  may  be  rarefied  and  brought  to  almost  any  degree  of  tenu- 
ity. This  is  accomplished  by  an  instrument  called  the  Air-Pumj)  or 
JExhausting  Syringe.  It  will  be  best  understood  by  describing  one 
of  the  simplest  kind.      It  consists,  essentially,  of 

1st.  A  Receiver  E,  or  chamber  from  which  the  exterior  air  is  ex- 
cluded, that  the  air  within  may  be  rarefied.  This  is  commonly  a 
bell-shaped  glass  vessel,  with  ground  edge,  over  which  a  small  quan- 
tity of  grease  is  smeared,  that  no  air  may  pass  through  any  remain- 


422 


ELEMENTS     OF    ANALYTICAL    MECHANICS. 


ing  inequalities   on  its  surfoce,  and  a  ground  glass  plate  m  n  imbedded 
in  a  metallic  table,  on  which  it  stands. 

2d.  A  Barrel  B, 
or  chamber  into 
which  the  air  in 
the  reservoir  is  to 
expand  itself.  It 
is  a  hollow  cylin- 
der of  metal  or 
glass,  connected 
with  the  receiver 
R  by  the  commu- 
nication ofg.     An 

air-tight  piston  P  is  made   to  move  back   and  forth  in   the  barrel  by 
means  of  the  handle  a. 

3d.  A  Stop-cock  h,  by  means  of  which  the  communication  between 
the  barrel  and  receiver  is  established  or  cut  off  at  pleasure.  This 
cock  is  a  conical  piece  of  metal  fitting  air-tight  into  an  aperture 
just  at  the  lower  end  of  the  barrel,  and  is  pierced  in  two  directions ; 
one  of  the  perforations  runs  transversely  through,  as  shown  in  the 
first  figure,  and  when  in  this  position  the  communication  between 
the  barrel  and  re- 
ceiver is  estab- 
lished ;  the  second 
perforation  passes 
in  the  direction  of  i:^ —       ^ 

I 


the  axis  from    the 

smaller    end,    and 

as     it     approaches 

the    first,  inclines    sideways,  and  runs   out   at   right    angles    to    it,   as 

indicated   in   the    second  figure.      In   this   position   of   the   cock,   the 

communication    between    the   receiver   and   barrel    is    cut    off",    whilst 

that  with  the  external  air  is  opened. 

Now,  suppose  the  piston  at  the  bottom  of  the  barrel,  and  the 
communication  between  the  barrel  and  the  receiver  established ; 
draw  the  piston  back,  the  air   in    the   receiver   will    rush   out   in   the 


APPLICATIONS.  4:23 

direction  indicated  by  the  arrow-head,  through  the  communication 
o/y,  into  the  vacant  space  within  the  barrel.  The  air  which  now 
occupies  both  the  barrel  and  receiver  is  less  dense  than  when  it  occu- 
pied the  receiver  alone.  Turn  the  cock  a  quarter  round,  the  com- 
munication between  the  receiver  and  barrel  is  cut  off,  and  that  be- 
tween the  latter  and  the  open  air  is  established;  push  the  piston  to 
the  bottom  of  the  barrel  again,  the  air  -within  the  barrel  will  be 
delivered  into  the  external  air.  Turn  the  cock  a  quarter  back,  the 
communication  between  the  barrel  and  receiver  is  restored ;  and 
the  same  operation  as  before  being  repeated,  a  certain  quantity  of 
air  will  be  transferred  from  the  receiver  to  the  exterior  space  at 
each  double  stroke  of  the  piston. 

To  find  the  degree  of  exhaustion  after  any  number  of  double 
strokes  of  the  piston,  denote  by  D  the  density  of  the  air  in  the  re- 
ceiver before  the  operation  begins,  being  the  same  as  that  of  the 
external  air ;  by  r  the  capacity  of  the  receiver,  by  b  that  of  the  bar- 
rel, and  by  2^  that  of  the  pipe.  At  the  beginning  of  the  operjvtion, 
the  piston  is  at  the  bottom  of  the  barrel,  and  the  internal  air  occu- 
pies the  receiver  and  pipe ;  when  the  jiiston  is  withdrawn  to  the 
opposite  end  of  the  barrel,  this  same  air  expands  and  occupies  the 
receiver,  pipe,  and  barrel ;  and  as  the  density  of  the  same  body  is 
inversely  proportional  to  the  space  it  occupies,  we  shall   have 

r  -{-  p  -{-  h     :     r+;?     ::     I)     :     x ; 

m  which  x  denotes  the  density  of  the  air  after  the  piston  is  drawn 
back  the  first  time.       From  this  proportion,  we  find 

X  =  D  • — -. 

r  +  p  +  6» 

The  cock  being  turned  a  quarter  round,  the  piston  pushed  back  to 
the  bottom  of  the  barrel,  and  the  cock  again  turned  to  open  the 
communication  with  the  receiver,  the  operation  is  repeated  upon  the 
air  whose  density  is  x,  and  wc  have 

r  +  p  +  b     :     r  +  p     :  :     i)  .  _!1±A^     :     x' ; 

r  +  p  +  b 

in  which  x'  is  the  density  after  the  second  backward  motion  -of  the 
piston,  or  after  the  second   double  stroke ;    and  we  find 


424 


ELEMENTS  OF  ANALYTICAL  MECHANICS. 


x'  =  D 


^r  +  p  +  b^ 

and  if  n  denote  the  number   of  double   strokes   of   the   piston,   and 
a:„  the  corresponding  density  of  the  remaining  air,  then  will 


D 


(  r  +  p  Y' 


^r  +  p  +  by 

From  which  it  is  obvious,  that  although  the  density  of  the  air  will 
become  less  and  less  at  every  double  stroke,  yet  it  can  never  be 
reduced  to  nothing,  however  great  n  may  be;  in  other  words,  the 
air  cannot  be  wholly  removed  from  the  receiver  by  the  air-pump. 
The  exhaustion  will  go  on  rapidly  in  proportion  as  the  barrel  is 
large  as  compared  with  the  receiver  and  pipe,  and  after  a  few  double 
strokes,  the  rarefaction  will  be  sufficient  for  all  practical  purposes. 
Suppose,  for  example,  the  receiver  to  contain  19  units  of  volume,  the 
pipe  1,  and   the  barrel  10;  then  will 

r  +  P       ^  ':^  ^  2  . 
r  +  p  +  b  ~  '60        ^' 

and   suppose  4   double   strokes   of  the  piston ;    then  will  «  =  4,  and 

r  +  p     \"        ,„..        16 


(    r+p    y  _ 


=  0,197,  nearly  ; 


f^^' 


<r  +  p  -\-  by  ^-^  81 

that  is,  after  4  double  strokes,  the  density  of  the  remaining  air  will 
be  but  about  two  tenths  of  the  original  density.  AVith  the  best 
machines,  the   air   may   be  rarefied  from   four  to   six  hundred  times. 

The   degree  of  rarefaction    is   indicated   in    a   very 
simple   manner   by    what    are    called   gauges .      These 
not   only    indicate    the    condition    of   the    air    in    the 
receiver,  but   also   warn    the    operator  of  any  leakage 
that  may  take  place  either  at  the  edge  of  the  receiver 
or    in    the  joints    of    the   instrument.      The    mode   in 
which  the  gauge  acts,  will  be  readily  understood  from 
the    discussion    of    the    barometer ;     it   will    be     suffi- 
cient here    simply  to  indicate  its  construction.      In  its  ^^  ^^ 
more  perfect  form,  it  consists  of  a  glass  tube,  about  60  inches  long, 
bent   in    the    middle    till    the    straight    portions    are    parallel    to    each 
other ;  one  end  is  closed,  and    the  branch  terminating  in   this    end  is 


APPLICATIONS. 


425 


filled  with  mercury.  A  scale  of  equal  parts  is  placed  between  the 
branches,  having  its  zero  at  a  point  midway  from  the  top  to  the 
bottom,  the  numbers  of  the  scale  increasing  in  both  directions.  It 
is  placed  so  that  the  branches  of  the  tube  shall  be  vertical,  with 
its  ends  upward,  and  inclosed  in  an  inverted  glass  vessel,  which 
communicates  with  the  receiver  of  the  air-pump. 

Eepeated  attempts  have  been  made  to  bring  the  air  pump  to 
still  higher  degrees  of  perfection  since  its  first  invention.  Self-acting 
valves,  opening  and  shutting  by  the  elastic  force  of  the  air,  have 
been  used  instead  of  cocks.  Two  barrels  have  been  employed  in- 
stead of  one,  so  that  an  uninterrupted  and  more  rapid  rarefaction 
of  the  air  is  brought  about,  the  piston  in  one  barrel  being  made 
to  ascend  while  that  of  the  other  descends.     The  most  serious  .defect 


was  that  by  which  a  portion  of  the  air  was  retained  between  the 
piston  and  the  bottom  of  the  barrel.  This  intervening  space  is  filled 
with   air    of    the   ordinary    density    at     each    descent   of    the    piston ; 


426     ELEMENTS  OF  ANALYTICAL  MECHANICS. 

when   the  cock   is  turned,    and  the   communication  re-established  with 
the  receiver,  this   air  forces   its    way   in    and   diminishes    the    rarefac- 
tion already  attained.      If  the  air    in   the   receiver  is   so   far    rarefied, 
that  one    stroke    of    the    piston    will    only   raise    such   a   quantity    as 
equals    the   air   contained   in    this    space,  it   is   plain    that    no    further 
exhaustion    can    be    effected   by    continuing   to    pump.      This   limit   to 
rarefaction   will    be    arrived    at    the    sooner,    in    proportion    as    the 
space  below    the    piston  is   larger;    and    one   chief  point    in    the   im- 
provements  has    been  to  diminish    this    space   as    much    as    possible. 
^^  is   a  highly  polished   cylinder  of  glass,  which  serves  as  the  bar- 
rel of  the  pump  ;  within  it  the  piston  works  perfectly  air-tight.     The 
piston   consists   of    washers   of    leather    soaked    in    oil,    or   of    cork 
covered  with  a  leather   cap,  and   tied   together   about  the   lower  end 
C  of  the    piston-rod   by    means    of   two    parallel    metal    plates.     The 
piston-rod   Cb,  which  is   toothed,  is  elevated  and  depressed  by  means 
of  a   cog-wheel  turned   by    the   handle  M.      If  a   thin  film  of  oil   be 
poured   upon    the   upper    surface   of    the   piston    the    friction    will    be 
lessened,  and  the  whole  will  be  rendered  more  air-tight.     To  diminish 
to   the   utmost  the  space   between  the   bottom  of  the   barrel   and  the 
piston-rod,  the  form  of  a   truncated   cone   is   given   to   the   latter,  so 
that  its    extremity  may  be  brought  as   nearly  as    possible    into   abso- 
lute  contact  with  the  cock  E;  this  space  is  therefore  rendered  indefi- 
nitely small,   the  oozing  of  the  oil  down  the  barrel  contributing  still 
further    to    lessen    it.      The   exchange-cock   E  has    the   double    bore 
already  described,  and  is   turned  by  a  short  lever,  to    which   motion 
is    communicated  by  a  rod   c  d.      The   communication    G  Jf  is  carried 
to  the  two  plates  /  and  K.   on  one  or  both   of  which  receivers  may 
be  placed  ;    the  two  cocks  JV  and   0  below  these  plates,  serve  to  cut 
off  the  rarefied   air  within   the  receivers  when    it   is  desired  to    leave 
them  for  any  length  of  time.      The  cock  0  is  also  an  exchange-cock, 
so  as  to  admit  the  external  air  into  the  receivers. 

Pumps  thus  constructed  have  advantages  over  such  as  work 
with  valves,  in  that  they  last  longer,  exhaust  better,  and  may  be 
employed  as  condensers  when  suitable  receivers  are  provided,  by 
merely  reversing  the  operations  of  the  exchange  valve  during  the 
motion  of  the  piston. 


TABLE    I. 

THE  TENACITIES   OF   DIFFERENT  SUBSTANCES,  AND   THE  RESISTANCES 
WHICH   THEY   OPPOSE  TO   DIRECT   COMPRESSION. 


SUBSTANCES    EXPERIMENTED   ON. 


Wrought-iron,  in  wire  from   l-20th  i 
to  l-30tli  of  an  inch  in  diame-  V 

tor ) 

in  wire,  1-lOth  of  an  inch  •     • 
in  bars,  Russian  (mean)  • 

English  (mean)    •     •     • 

hammered 

rolled  in  sheets,  and  cut  length-  | 

wise ) 

ditto,  cut  crosswise    • 
in  chains,  oval  links  6  in.  clear,  | 
iron  U  in.  diameter  •    •     •     •  f 
ditto,  Bruutou's,  with  stay  across  | 

link i 

Cast  Iron,  quality  No.  1      .     •     •     • 


Steel,  cast 

cast  and  tilted  •  •  •  • 
blistered  and  hammered  • 
shear  


Damascus 

ditto,  once  refined 
ditto,  twice  refined 
Copper,  east  .... 

hammered    •     • 

sheet 

wire 

Platinum  wire     • 
Silver,  cast     .... 

wire 

Gold,  cast 

wire 

Brass,  yellow  (tine)  •  • 
Gun  metal  (hard)  •  • 
Tin,  cast    

wire 

Lead,  cast      •     •     •     • 

milled  sheet 

wire 


60  to  91 

36  to  43 
27 

3o 

14 
18 
2Ij 


25 

to  qi 
to  8 
to  gi 
44 
60 
59J 

57 
5o 
3i 
36 
44 
8^ 
i5 
21 
27J 
17 
i3 

17 
9 
14 


Lame 


Telford 
Lame 


Brunei 
Mitis 


Brown 
Barlow 


Hod2:kinson 


Mitis 
Rennie 


Mitis 

Rennie 
Kingston 
Guvton, 


8 
16 

Rennie 

73 

2 
3 

4-5ths 
i4 
1,1 

Tredgold 
Guyton 

7 
3. 

c  —     . 


38  .to  41 
37  to  48 
5i    to  65 


52 
46 


H5 
=  g 

Is 


Hodgkinson 


Rennie 


*The  strongest  quality  of  cast  iron,  is  a  Scotch  iron  known  as  the  Devon  Hot  Blast,  No.  3:  its  tenaci- 
ty is  9J  tons  per  square  inch,  and  its  resistance  to  compression  65  tons.  The  experiments  of  Major 
Wade  on  the  gun  iron  at  West  Point  Foundry,  and  at  Boston,  give  results  as  high  as  10  to  16  tons,  and 
on  small  cast  bars,  as  high  as  17  tons.— See  Ordnance  Manual,  1850,  p.  402. 


TABLE    I. 

TABLE  I — continued. 


420 


SUBSTANCES     EXPERIMENTED    ON. 


Stone,  slate  (Welsli)      •     • 

Marble  (white)  •     •     • 

Giviy 

Portland 

Craigleith  freestone     • 

Braiiiley  Fall  sandstone 

Cornisli  granite 

Peterhead  ditto 

Limestone  (compact  blk) 

Purbeck 

Aberdeen  granite   •     • 
Brick,  pale  red    .... 

red 

Hammersmith  (pavior's) 
ditto      (burnt)  •     • 

Chalk 

Plaster  of  Paris  .... 

Glass,  plate 

Bone  (ox) 

Hemp  fibres  glued  together 
Strips  of  paper  glued  together 
Wood,  Box,  spec,  gravity 

Ash     • 

Teak.       • 

Beech 

Ouk     .     . 

Ditto  •     • 

Fir      •     • 

Pear    •     • 

Mahogany 

Elm     •     • 

Pine,  American 

Deal,  wliite   • 


,862 

,9 

,1 

,9'^ 

,77 

,6 

,646 

,637 


5,7 
4 


,o3 

4 

2,2 
41 
i3 

9 
8 


Barlow 


1,4 

1,6 
2.4 
2.7 

2.8 
3,7 
4 
4 
5 
,56 

,B 
I 

1,4 
,22 


J,7 


,57 

,73 
,86 


Kcnuie 


430 


TABLE    II. 


TABLE  n. 


OF  THE  DENSITIES  AND  VOLUMES  OF  WATEE  AT  DIFFEEENT  DEGEEES 
OF  HEAT,  (ACCORDING  TO  STAMPFEE),  FOR  EVERY  2i  DEGREES  OF 
FAHRENHEIT'S  SCALE. 

(Jahrbuch  des  Polylechnischen  Institutes  in   fVcin,  Bd.  16,  S.  70). 


Temperature. 

Density. 

Diir. 

V 
Volume. 

Diff. 

0 

32,00 

0,999887 

1,0001 13 

34,23 

0,999950 

63 

I,oooo5o 

63 

36,5o 

0,999988 

38 

1,000012 

38 

38,75 

1 ,000000 

12 

1,000000 

12 

4i,oo 

0,999988 

12 

1,000012 

12 

43.23 

0,999952 

35 

1 ,000047 

35 

45;5o 

0,999894 

58 

1,000106 

«9 

47,75 

0,999813 

81 

1,000187 

81 

5o.oo 

0,999711 

102 

1,000289 

102 

52,25 

0,999587 

124 

1, 0004 1 3 

124 

54, 5o 

0,999442 

145 

i,ooo558 

145 

56,75 

0,999278 

164 

1,000723 

i65 

5q,oo 

0,999095 

i83 

1 ,000906 

1 83 

61.25 

0,998893 

202 

1,001108 

202 

63,5o 

0,998673 

220 

1,001329 

221 

65,75 

0,998435 

238 

1,001567 

238 

68,00 

0,998180 

255 

1,001822 

255 

70,25 

0,99-909 

271 

i,oo2og5 

273 

72,50 

0.997622 

287 

1,002384 

289 

74,75 

0,997320 

3o2 

1,002687 

3o3 

77,00 

0,997003 

3.7 

i,oo3oo5 

3i8 

79,25 

0.996673 

33o 

i,oo3338 

333 

81, 5o 

0,996329 

344 

1, 003685 

347 

83,75 

0,995971 

358 

1,004045 

36o 

86,00 

0,993601 

370 

1,004418 

373 

88,25 

0,995219 

382 

1,004804 

386 

90,50 

0,994825 

394 

I,005202 

398 

92,75 

0.994420 

40  5 

i,oo56i2 

410 

95,00 

0,994004 

416 

I,oo6o32 

420 

97,25 

0.993579 

425 

1,006462 

43o 

99.50 

0,993145 

434 

I, 006902 

440 

With  this  tiible  it  is  easy  to  tind  the  specific  gravity  by  means  of  water  at  any  teinperalure. 
Suppose,  for  example,  the  specific  gravity  S'  in  Equation  (456),  had  lieen  found  at  the  tempera- 
ture of  590,  then  would  D,,  in  that  equation  be  O.QOOOO.i,  and  the  specific  gravity  of  the  body 
referred   to  water  at  its    greatest  density,  would  be  given  by 

S  =  S'  X  0,999095. 


TABLE    III. 


431 


TABLE  III. 


OF  THE  SPECIFIC  GKAVITIilS  OF  SOME  OF  THE  MOST  IMPORTANT  BODIES. 
[The   ileiisily  of  ilislillcd   water   is   reckoned  in   this   Table   at   its   iiiaxijiuiiii   38J0  F.  =  1,000]. 


Name  of  the  Body. 


Specific  GfHvity. 


I.    SOLID  BODIES. 

(1)  Metals. 


Antimony  (of  the  laboratory) 

Brass       •  ... 

Bronze  for  cannon,  according 

Ditto,  mean     • 

Clipper,  melted 

Ditto,  hammered 

Ditto,  wire-drawn    • 

(lold,  melted    • 

Ditto,  hammered 

Iron,  \vrought 

Ditto,  cast,  a  mean  • 

Ditto,  gray 

Ditto,  white    • 

Ditto  for  cannon,  a  mean 

Lead,  pure  melted  • 

Ditto,  flattened 

Platinum,  native 

Ditto,  melted   • 

Ditto,  hammered  and  wire-dn 

Quicksilver,  at  32°  Fahr. 

Silver,  pure  melted 

Ditto,  liammcred 

Steel,  cast 

Ditto,  wrought 

Ditto,  much  liardened 

Ditto,  slightly 

Tin,  clietnically  pure 

Ditto,  hammered 

Ditto,  Bohemian  and  Sa.\on 

Ditto,  English 

Zinc,  melted    • 

Ditto,  rolled    • 


to  Li 


(2)  Building  Stones 


Alabiister 

Basalt      .... 

Dolerite  .... 

Gneiss     .... 

Granite    .... 

Hornblende 

Liniestonc,  various  kinds 

Plionolitc 

Por[ili\ry 

Quartz     .... 

Sandstone,  various  kinds,  a  me 

Stones  for  building  • 

Syenite    .... 

Trachyte 

Brick       .... 


ut.  Matzka 


4,2           — 

4,7 

7,6         - 

8,8 

8,414    - 

8,974 

8,758 

7,788     - 

8,726 

8,878     — 

8,9 

8,78 

19,238     — 

19,253 

19,361     — 

19,6 

7,207     — 

7,788 

7,25i 

7,2 

7,5 

7,21         — 

7,3o 

ii.33o3 

11,388 

16,0        — 

18,94 

20,855 

21,25 

i3,568    — 

13,598 

10.474 

10, 5i       — 

10,622 

7,9'9 

7,840 

7,818 

7,833 

7,291 

7,299    — 

7,475 

7,3i2 

7,291 

6,861     — 

7,2i5 

7,191 

2,7         — 

3,0 

2,«           — . 

3,1 

2.72         — 

2,93 

2,5           — 

2,9 

2,5           — 

2,66 

2,9      — 

3,1 

2,64    - 

2,72 

2,5l         — 

2,69 

2,4           — 

2,6   •• 

2,56      — 

2.75 

2.2           — 

2,5 

1,66       — 

2,62 

2.5           — 

3. 

2,4          — 

2.6 

1,41     — 

1.86 

432 


TABLE    III. 


TABLE   lll—Co7itmuecl 


Name  of  the  Body. 


Specific  Gravity. 


I.    SOLID  BODIES. 
(3)  Woods. 

Alder      

Ash 

Aspen     •         

Birch 

Box 

Elm 

Fir 

Hornbeam        ...... 

Horse-chestnut 

Larch      ....... 

Lime       ....... 

Maple 

Oak 

Ditto,  another  specimen  .... 
Pine,  Pinus  Abies  Picea  .  .  .  • 
Ditto,  Pimts  Sylvestris    .         .         •         • 

Poplar  (Italian) 

"VVillow  ..••••• 
Ditto,  white 

(4)  Various  Solid  Bodies. 

Charcoal,  of  cork 

Ditto,  soft  wood 

Ditto,  oak 

Coal         ....... 

Coke 

Earth,  common 

rough  sand         ..... 

rough  earth,  with  gravel    • 

moist  sand 

gravelly  soil 

clay 

clay  or  loam,  with  gravel  • 

Flint,  dark 

Ditto,  white 

Gunpowder,  loosely  filled  in 

coarse  powder  .         .         .         •         • 

musket  ditto     • 
Ditto,  slightly  shaken  down 

musket-powder         .... 

Ditto,  solid 

Ice 

Lime,  unslacked  ..... 
Eesin,  common        ..... 

Eoek-salt 

Saltpetre,  melted 

Ditto,  crystallized  .  .  ■  •  • 
Slate-pencil  ....•• 
'Sulphur  ....••• 
Tallow  ....••• 
Turpentine  ....•• 
Wax,  white     ...... 

Ditto,  yellow 

Ditto,  shoemaker's 


Fresh-fplled. 

Drv. 

0,8571 

OjDOOI 

o,go36 

0,6440 

0,7654 

o,43o2 

0,9012 

0,6274 

0,9822 

0,3907 

0.9476 

0,5474 

0,8941 

o,555o 

0,9452 

0,7695 

0,8614 

0,5749 

0,9206 

0,4735 

0,8170 

0,4390 

o,9o36 

0,6092 

1,0494 

0,6777 

1,0754 

0,7075 

0,8699 

0,4716 

0,9121 

0,3302 

0,7634 

0,3931 

0,7 1 55 

0,5289 
0,4873 

0,9859 

0,1 

0,28 
1,573 

1,232 

1,865 

1,48 

1,92 

2,02 

2,o5 

2,07 

2,l5 

2,48 

2,542 
2,741 

0,886 
0,992 


2,248    — 
0,916    — 

1,842 
1,089 

2,237 
2,745 
1,900 

1,8        — 

1,92      — 

0,942 

0,991 

0,969 

0,063 

0,897 


0,44 
i,5io 


2,563 
0.9268 


2,24 
1.99 


TABLE    III. 


433 


TABLE   111— Continued. 


Name  of  the  Body. 

Specific  Gravity. 

11.  LIQUIDS. 

Acid,  acetic 

i,o63 

Ditto,  inuriatic 

1,211 

Ditto,  nitric,  oonccntrated 

1,521      —      1,522 

J)itto,  liulpliuric,  Ensflish 

1,845 

Ditto,  concentrated  (Nordh.) 

i,86o 

Alcohol,  free  from  -n 

'ater  • 

0,792 

Ditto,  Common 

0,824    —     0,79 

Aninupniac,  liquid 

0,875 

Aquafortis,  double 

i,3oo 

Ditto,  sindc     • 

1,200 

Beer         •         • 

1,023     —     i,o34 

Etjier,  acetic    • 

0.866 

Ditto,  muriatic 

0,845    —    0.874 

Ditto,  nitric     • 

0,886 

Ditto,  sulphuric 

0,715 

Oil,  linseed      • 

0.928    —    0,953 

Ditto,  (ilivc      • 

0.9 1 5 

Ditto,  turpentiue 

0,792    —    0,891 

Ditto,  wluile    • 

0,923 

Quicksilver 

13.568    —  13,598 

Water,  distilled 

1,000 

Ditto,  rain 

i,ooi3 

Ditto,  sea 

1,0265  —     1,028 

Wine 

• 

0,992    —     i,o38 

III.  GASES. 

Briroinelei 

Wiiter=  1. 

3U  !ii. 

■- 

Teiii;i.  38JO  F. 

■l"<iii.=3'2^ 

Atmospheric  air  =  jj-q  ^^    ' 

0,00 1 3o 

1,0000 

Carbonic  acid  gas 

0,00198 

1,5240 

(,'arbonic  oxide  gas   •          •      _    • 

0,00126 

0,9569 

Ca'-bureted  hydrogen,  a  maximum 

0,00127 

0^9784 

Ditto,  from  Coals      • 

i 
( 

0,00039 
0, 00085 

o.3ooo 
0,5596 

Chlorine  .... 

o,oo32l 

2,4700 

Hydriodic  gas  • 

0,00577 

4,443o 

Hydrogen 

0,0000895 

0,0688 

llydroMilphurie  acid  gas  • 

o.ooi55 

1,1912 

Muriatic  acid  gas 

0  00162 

1,2474 

Nitrogen          ■         ... 

0,00127 

0,9760 

Oxygen    .... 

0,00143 

1,1026 

Phosphureted  hydrogen  gas 

o,ooii3 

0,8700 

Steam  at  212°  Falir. 

o,'ooo82 

0.6235 

Sulphurous  acid  gas 

0,00292 

2,2470 

28 


434: 


TABLE  IV. 


TABLE  IV. 

TABLE  FOE  FINDING  ALTITUDES. 


netached  Thermomeier. 

h+l' 

A 

0  +  t' 

A 

t/+«' 

A 

t,  +  t' 

A 

40 

4,7689067 

75 

4,7859208 

no 

4,8022936 

145 

4,8180714 

41 

,7694021 

76 

,7863973 

III 

,8027525 

146 

,8i85i4o 

42 

,7698971 

77 

,7868733 

112 

,8032109 

'^J 

,8189559 

43 

,7703911 

78 

,7873487 

ii3 

,8086687 

148 

,8198975 

44 

,77o885i 

79 

,7878236 

114 

,8041261 

149 

,8198887 

43 

,7713785 

80 

,7882979 

n5 

,8o4583o 

i5o 

,8202794 

46 

,7718711 

81 

,7887719 

116 

,8050893 

i5i 

,8207196 

47 

,7723633 

82 

,7892451 

117 

,8054953 

l52 

,8211594 

48 

,7728548 

83 

,7897180 

118 

,8059309 

1 53 

,8215988 

49 

,7733457 

84 

,7901903 

119 

,8o64o58 

i54 

,8220377 

5o 

,7738363 

85 

,7906621 

120 

,8068604 

1 55 

,8224761 

5i 

,7743261 

86 

,7911335 

121 

,8078144 

1 56 

,8229141 

52 

,77-iSi53 

87 

,7916042 

122 

,8077680 

i57 

,8283517 

53 

,7753042 

88 

,7920745 

123 

,8082211 

1 58 

,8287888 

54 

,7757925 

89 

,7925441 

124 

,8086787 

159 

,8242256 

55 

,7762802 

90 

,793oi35 

125 

,8091238 

160 

,8246618 

56 

,7767674 

9' 

,7934822 

126 

,8095776 

161 

,8230976 

57 

,7772540 

92 

,7939504 

127 

,8100287 

162 

,8255881 

58 

,7777400 

93 

,7944182 

128 

,8104793 

168 

,8259680 

59 

,7782256 

94 

,7948854 

129 

,8109298 

164 

,8264024 

60 

,7787105 

95 

,7953521 

i3o 

,8118796 

i65 

,8268365 

61 

,7791949 

96 

,7958184 

i3i 

,8118290 

166 

,8272701 

62 

,7796788 

97 

,7962841 

l32 

,8122778 

167 

,8277084 

63 

,7801622 

98 

,7967493 

i33 

,8127268 

168 

,8281862 

64 

,7806450 

99 

,7972141 

i34 

,8181742 

169 

,8285685 

65 

,7811272 

100 

,7976784 

i35 

,8186216 

170 

.8290005 

66 

,7816090 

101 

,7981421 

1 36 

,8140688 

171 

,8294819 

67 

,7820902 

102 

,7986054 

137 

,81451 53 

172 

,8298629 

68 

.  ,7825709 

io3 

,7990681 

i38 

,8149614 

173 

,8802987 

69 

,783o5u 

104 

,7995303 

139 

,8154070 

174 

,8807288 

70 

,7835306 

io5 

,7999921 

140 

,8i58523 

175 

,88 11 586 

7' 

,7840098 

106 

,8004533 

141 

,8162970 

176 

,88i583o 

72 

,7844883 

107 

,8009142 

142 

,8167418 

177 

,8820119 

73 

,7849664 

108 

,8018744 

143 

,8171852 

178 

,8824404 

7^ 

4,7854438 

109 

4,8018343 

144 

4,8176285 

179 

4,8828686 

TABLE   IV. 


435 


TABLE  IV— continued. 

"WITH  THE  BAROMETEE. 


Latitude. 

Attached  Thermometer. 

yy 

B 

T-T' 

C 

c 

0° 

3 
6 
9 

13 

i5 

i8 

21 

24 

27 

3o 
33 
36 
39 

42 

45 
48 

5o 
5i 

52 

53 
54 
55 
56 

57 
58 
59 
6o 
63 
66 

90 

0,0011689 
,0011624 
,0011433 
,0011117 
,0010679 
,0010124 
,0009439 
,0008689 
,0007823 
,0006874 
,0005848 
,0004758 
, 00036 1 5 
,0002433 

,0001223 

,0000000 
9,9998773 

,9998372 

■999707 
,9997366 

,9997 '67 
,9996772 
,9996381 
,9993993 
.   ,9993613 
,9993237 
,9994866 
,9994502 
,9994144 
,9993113 
,9992161 
,9991293 
,9989^^52 
,9988854 
9,9988300 

0'^ 

I 
2 

3 
4 
5 
6 

7 
8 

9 
10 
II 
12 
i3 
14 
i5 
16 

17 
18 

•9 

20 
21 
22 

23 

24 

25 

26 

27 
28 

29 

3o 
3i 

+ 
0,0000000 
,0000434 
,0000869 
,oooi3o3 
,0001737 
,0002171 

,O0026o5 

,ooo3o39 
,0003473 
,0003907 
,0004341 
,0004775 
,0003208 
,0003642 
,0006076 
,ooo65io 
,0006043 
,0007377 
,0007810 
,0008244 
,0008677 
,0009111 
,0009344 
,0009977 
,0010411 
,0010844 
,0011277 
,0011710 
,0012143 
,0013576 
,00 1 3009 
0,0013442 

0,0000000 
9,9999566 
,99991 3 1 
,9998697 
,9998262 
,9997828 
,9997393 
,9996939 
,9996524 
,9996090 
,9995655 
,9995220 
,9994785 
,9994330 
,9993916 
,9993481 
,9993046 
,999261 1 
,9992176 
,9991741 
,999i3o5 
,9990870 
,9990435 
,9900000 
,9989564 
,9989129 
,9988694 
,9988238 
,9987823 
,9987387 
,9986q52 
9,9986316 

i 


436 


TABLE    V. 


TABLE  Y. 

COEFFICIENT  VALUES,  FOE  THE  DISCIIAKGE  OF  FLUIDS  THEOUGII  THIN 
PLATES,  THE  OKIFICES  BEING  EE.MOTE  FEOM  THE  LATEEAL  FACES 
OF  THE  VESSEL. 


Headoffluiri 

ahove  the 

centre  of  the 

orilice,  in  feet. 

Values  of  the  coefticients  for  orifices  whose  s:iiallest  dimensions  or 
diameters  are— 

ft- 
0,66 

0*33 

ft- 
0,16 

0,08 

ft- 
0,07 

o,o3 

o,o5 
0,07 
o,i3 
0,20 
0,26 
0,33 
0,66 
1,00 
1,64 
3,28 
5,00 
6,65 
32,75 

o,5g3 
0,596 
0,601 
0,602 
o,6o5 
o,6o3 
0,602 
0,600 

0,592 
0,602 
o,6oS 
o,6i3 
0,617 
0,617 
(3,6i5 
0,612 
0,610 
0,600 

0,618 
0,620 
0.625 
o,6Jo 
0,63 1 
o,63o 
0,628 
0,626 
0,620 
0,61 5 
0,600 

0,627 
0,632 
0,640 
0,638 
0,637 
0,634 
0,632 
o,63o 
0,628 
0,620 
0,61 5 
0,600 

0,660 

0,657 
0,656 
0,655 
0,655 
0,634 
0,644 
0,640 
0,633 
0,631 
0,610 
0,600 

0,700 
0,696 
0,685 
0,677 
0,672 
0,667 
0,655 
o,65o 
0,644 
0,632 
0,618 
0,610 
0,600 

In  the  instance  of  gas,  the  generating  head  is  always  ^renter  than  G,t>j  ft.,  and  the  coefficient  0,C, 
or  0,61,  is  taken  in  all  cases. 

For  orifices  larger  than  0,66  ft.,  the  coefficients  are  taken  as  for  this  dimension ;  for  orifices  smaller 
than  0,03  ft.,  the  coethcient.s  are  the  same  as  ibr  this  btter;  finally,  for  orifices  between  those  of  the 
tahle,  we  take  coefficients  whose  values  are  a  mean  between  the  latter,  corresponding  to  the  given  head. 


t 


TABLE   VI. 


437 


TABLE  VI. 

EXPERIMENTS  ON  FKICTION,  WITHOUT  UNGUENTS.    BY  M.  MOEIN. 

The  surfaces  of  friction  were  viiried  from  q,o3336  to  2,7987  square  feet,  tbe  pressures  from 
88  lbs.  to  22o5  lbs.,  aud  tlje  velocities  from  a  scarcely  perceptible  motion  to  9,84  feet  per 
second.  The  surfaces  of  wood  were  planed,  and  those  of  metal  filed  and  polished  with  the 
greatest  care,  and  carefully  wiped  after  every  c.\iieriment.  The  presence  of  unguents  was 
especially  guarded  against. 


Friction  of 

Friction  or 

SURFACES  OF  CONTACT. 

MOTIO.N.* 

QUIKSCBNCK.f 

1 

ti'- 

g 

a 

<B-^ 

;=  ^  ■^ 

Q'- 

,4<;cs  . 

0  0 

^<K 

Oak  upon  oak,  the  direction  of  the  fibres  ) 
being  parallel  to  the  motion      •     •     •  ) 

0,478 

25" 

33' 

0,625 

32«>      I' 

Oak  upon  oak,  the  directions  of  the  fibres  ] 
of  the  moving  surface  being  perpcn-  1 
dicular  to  those  of  the  quiescent  sur-  | 

0,824 

17 

58 

0,540 

23      23 

face  and  to  the  direction  of  the  motionj  J 

Oak  upon  oak,  the  fibres  of  the  both  sur-  1 

faces  being  perpendicular  to  the  direc-  v 

0,336 

18 

35 

tion  of  the  motion ) 

Oak  upon  oak,  the  fibres  of  the  moving  ] 

surface  being  perpendicular  t>)  tlie  sur- 

face of  contact,  and  those  of  the  surface  I 

0,192 

10 

52 

0,271 

i5     10 

at  rest  parallel  to  the  direction  of  the 

motion J 

Oak  upon  oak,  the  fibres  of  both  surfaces  | 

being  perpendicular  to  the  surface  of  V 

0,43 

23     17 

contact,  or  the  pieces  end  to  end   •      •  \ 

Elm  upon  oak,  the  direction  of  the  fibres  1 
being  parallel  to  the  motion      •     •     •  ) 

0,432 

23 

22 

0,694 

34    46 

Oak  upon  elm,  ditto§ 

0,246 

i3 

5o 

0,376 

20    37 

Elm  upon  oak,  the  fibres  of  the  moving  ^ 

surface  (the  elm)  being  perpendicular  to  ! 

0,430 

24 

16 

0,570 

29    41 

those  of  the  quiescent  surface  (the  oak)  [ 

and  to  the  direction  of  the  motion-     •  J 

Ash  upon  oak,  the  fibres  of  both  surfaces  1 

being  parallel  to  the  direction  of  the  >■ 

0,400 

21 

49 

0,570 

29    41 

motion ) 

Fir  upon  oak,  the  fibres  of  both  surfaces  j 

being  parallel  to  the  direction  of  the  >■ 

0.355 

19 

33 

0,520 

27    29 

motion ) 

Beach  upon  oak,  ditto 

o,36o 

'9 

48 

0,53 

27     56 

Wild  pear-tree  upon  oak,  ditto    • 

0,370 

20 

'9 

0.440 

23    45 

Service-tree  upon  oak,  ditto    .... 

0,400 

21 

49 

0.570 

29    41 

W'rought  iron  upon  oak,  ditto]]  • 

0,619 

3i 

47 

0,619 

3i     47 

•  The  friction  in  this  case  varies  but  very  slialilly  from  the  inenn. 

t  Tlie  friction  in  this  c;ise  varies  considerably  from  the  nienn.  In  all  the  experiments  the  surfaces 
had  !)een  15  minutes  in  contact. 

i  The  dimensions  of  the  surfaces  of  contact  were  in  this  o.xperiment  ,947  square  feet,  unci  the  resulU 
were  nearly  uniform.  When  the  dimensions  were  diminished  to  ,043.  a  learinp  of  the  fibre  became  appa- 
rent in  the  case  of  motion,  and  there  were  symptoms  of  the  combustion  of  the  wood  ;  from  these  cir- 
cumstances there  resulted  an  irregularity  in  tlie  friction  indicative  of  excessive  pressure. 

^  It  is  worthy  of  remark  that  the  friction  of  oak  U|Hm  elm  is  but  five-ninths  of  thnt  of  elm  upon  oak. 

11  In  the  experiments  in  which  f)neof  the  snrfaces  was  of  metal,  small  particles  of  the  metal  began, 
after  a  time,  to  be  apparent  upon  the  wooil,  givinj;  It  a  polished  metallic  appearance  ;  these  were  at  every 
experiment  wiped  off;  they  indicated  a  wearing  of  the  metal.  The  friction  of  motion  and  that  of  quies- 
cence, in  these  experiments,  coincided.    The  results  were  remarkably  uniform. 


'138 


TABLE    VI. 


TABLE  YI — continued. 


SURFACES  OF   CONTACT. 


Wrcu2:ht  iron  upon  oak,  the  surfaces  ) 
being  greased  and  well  wetted-     •      •  j 

Wrought  iron  upon  elm     •      •      •    _■ 

Wrought  iron  upon  cast  iron,  the  fibres  j 
of  the  iron  being  parallel  to  the  motion  f 

Wrought  iron  upon  wrought  iron,  the  ) 
fibres  of  both  surfaces  being  parallel  /■ 
to  the  motion  •     •  " ) 

Cast  iron  upon  oak,  ditto 

Ditto,  the  surfaces  being  greased  and  \ 
wetted ) 

Cast  iron  upon  elm  • 

Cast  iron  upon  cast  iron 

Ditto,  water  being  interposed  between  ) 
the  surfaces ( 

Cast  iron  upon  brass 

Oak  upon  cast  iron,  the  fibres  of  the  wood  i 
being  perpendicular  to  the  dii-ection  > 
of  the  motion ) 

Hornbeam  upon  cast  iron — fibres  paral-  i 
lei  to  motion ) 

Wild  pear-tree  upon  cast  iron — fibres  I 
parallel  to  the  motion ) 

Steel  upon  east  iron 

Steel  upon  brass      •     • 

Yellow  copper  upon  cast  iron  •     •     •     • 
Ditto  oak      .... 

Brass  upon  cast  iron 

Brass  upon  wrought  iron,  the  fibres  of ) 
the  iron  being  parallel  to  the  motion   •  ) 

Wrought  iron  upon  brass 

Brass  upon  brass 

Black  leather  (curried)  upon  oak*    • 

Ox  hide  (such  as  that  used  for  soles  and 
for  the  stuffing  of  pistons)  upon  oak, 

rough     

Ditto        ditto        ditto    smooth     • 

Leather  as  above,  polished  and  hardened 
by  hammering 

Hempen   girth,  or  pulley-band,  (sangle  "j 
de  ch.anvre,)  upon  oak,   the  fibres   of  ' 
the  wood  and  the  direction  of  the  cord 
being  parallel  to  the  motion     • 

Hempen  matting,  woven  with  small 
cords,  ditto. •      • 

Old  cordage,  \\  inch  in  diameter,  ditto+ 


Friction  or 
Motion. 


0,256 

0,232 

0,194 

0,1 33 

0,490 

0,195 

0,1  52 

o.3i4 
0,147 

0,372 

0,394 

0,436 
0,202 

0,l52 

0,189 
0,617 
0.217 

0,161 

0,172 
0,201 
0.265 

0,52 

0,335 
o,2g6 

0,52 

0,32 

0,52 


14°  22' 

14  9 

10  59 

7  52 

26  7 

11  3 

8  39 

17  26 
8  22 

20  25 

21  3i 

23  34 

11  26 

8  39 

10  49 
3i  41 

12  i5 

9  9 
9  46 

11  22 
14  5i 

27  29 

18  3i 

16  3o 

27  29 

17  45 
27  29 


Friction  of 

ClUIESCENCK. 


C  3 


0,649 

0,194 

0,1 37 

0,646 
0,162 


0,617 


0,74 

o,6o5 

0,43 

0,64 

o,5o 
0,79 


33°     0' 

10     59 

7     49 

32      52 

9     i3 


3i    41 


36  3i 

3i  u 

23  17 

32  38 

26  34 

38  19 


•  The  friction  of  motion  was  very  nearly  the  same  whether  the  surface  of  cont;ict  was  the  inside 
or  the  outsi.ie  of  the  skin.— The  constancy  of  the  coefficient  of  the  friction  of  motion  was  equally  ap- 
parent in  the  rough  and  the  smooth  skins. 

t  All  the  above  e.xperiments,  except  that  with  curried  black  leather,  presented  the  phenomenon  of 
n  change  in  the  polish  of  the  surfaces  of  friction-a  state  of  their  surfaces  necessary  to,  and  dependent 
upon,  their  motion  upon  one  another. 


TABLE   TI. 

TABLE  VI — contimted. 


439 


SURFACES  or  CONTACT. 


Calcareous  oolitic  stone,  used  in  building,  "1 
of  a  moderately  hard  qualitj-,  called  [ 
stone   of    Jauuionl — upon    the  same  ( 

stone J 

Hard  calcareous  stone  of  Brouck,  of  a  I 
light  gray  color,  suscejitible  of  taking  ! 
a  line  polish,  (the  inu>ciielkalk,)  niov-  [ 

ing  upon  the  r^anie  stone j 

The  soft  stone  mentioned  above,  upon  f_ 

the  hard ) 

The  hard  stone  mentioned  above  upon 

the  soft 

Common  brick  upon  the  stone  of  Jaumotit 
Oak  upon  ditto,  the  tibres  of  the  wood  ) 
being  perpendicidar  to  the  surface  of  r 

the  stone ) 

Wrought  iron  upon  ditto,  ditto    • 
Common  brick  upon  tlie  stone  of  Brouclc 
Oak  as  before  (endwise)  upon  ditto  • 
Iron,  ditto  ditto 


Friction  of 
Motion. 


1^ 


0,64 


0,38 

0,65 

0,67 
0,65 

0,38 

0,69 
0,60 
o38 
0,24 


c  =   i 


32^  38' 


20  49 

33  2 

33  5o 

33  2 

20  49 

34  37 
3o  58 
20  49 
i3  3o 


Friction  of 
quikscknck. 


0,74 


0,70 


0,75 
0,73 

0,65 

0,63 

0.49 
0,67 
0,64 
0,42 


36"  3i' 


35 


36  53 

36  53 

33  2 

32  i3 

26  7 

33  5o 
32  33 
22  47 


440 


TABLE  VII. 


TABLE  VII. 

EXPEEIMENTS  ON  THE  FEICTION  OF  UNCTUOUS  SUEFACES. 
BY  M.  MOEIN. 

la  these  experiments  the  surfaces,  after  having  been  smeared  with  an  unguent,  were 
•wiped,  so  that  no  interposing  layer  of  the  unguent  prevented  their  intimate  contact. 


Friction  of 

Friction  of          | 

SURFACES  OF  CONTACT. 

Motion. 

QUIE 

5CENCE. 

*^  d 

» 

..-  c 

0 

c  c 

0 

CO 

"^  H 

S° 

c 

fct  2    £3 

«£ 

£  ^ 

•i 

c  • 

■5  "ei-S 
.SCO 

O'o 

l^< 

- 

c  = 

J<Pi 

Oak  upon  oak,  the  fibres  being  parallel  to  ) 
the  motion ) 

o,io3 

6° 

10' 

0,390 

21°     19' 

Ditto,  the  fibres  of  the  moving  body  be-  ( 

0,143 

8 

0  3i4 

17      26 

ing  perpendicular  to  the  motion-         •  \ 

9 

Oak  upon  elm,  fibres  parallel- 

0,1 36 

7 

45 

Elm  upon  oak,  ditto      .... 

0,1 19 

6 

48 

0,420 

22      47 

Beecli  upon  oak,  ditto  ... 

o,33o 

j8 

16 

Elm  upon  elm,  ditto      •         • 

0,140 

7 

^ 

Wrought  iron  upon  elm,  ditto 

o,i38 

7 

52 

Ditto  upon  wrought  iron,  ditto 

0)i77 

10 

3 

Ditto  upon  cast  iron,  ditto     •         • 

0,118 

6    44 

Cast  iron  upon  wrought  iron,  ditto 

0,143 

8 

9 

Wrought  iron  upon  brass,  ditto     • 

0,160 

9 

6 

Brass  upon  wrought  iron 

0,166 

9 

26 

Cast  iron  upon  oak,  ditto 

0,107 

6 

7 

0,100 

5    43 

Ditto  upon  elm,  ditto,  the  unguent  being 
tallow 

0,123 

7 

8 

Ditto,   ditto,  the  unguent  being  hog's 
lard  and  black  lead     .         .         .         ■ 

0,187 

7 

49 

Elm  upon  cast  iron,  fibres  parallel  • 

o,i35 

7 

42 

0,098 

5    36 

Cast  iron  upon  east  iron 

0,144 

8 

12 

Ditto  upon  brass 

0,l32 

7 

32 

Brass  upon  cast  iron     .... 

0,107 

6 

7 

Ditto  upon  brass 

o,i34 

7 

38 

0,164 

9     19 

Copper  upon  oak  •         •         • 
Yellow  copper  upon  cast  iron 

0,100 

5 

43 

o,ii5 

6 

34 

Leather  (ox  hide)  well  tanned  upon  cast  1 
iron,  wetted ) 

0,229 

12 

54 

0,267 

14    57 

Ditto  upon  brass,  wetted 

0,244 

i3 

43 

TABLE    VIII. 

TABLE  Vm. 


Ui 


EXPERIMENTS  ON  FKICTION  WITH  UNGUENTS  INTERPOSED.    BY  M.  MOKIN. 

The  e.xtent  of  the  surfaces  in  these  e.\periments  bore  such  a  relation  to  the  pre.s«urc,  a* 
to  cause  them  to  be  separated  from  one  anotlier  throughout  by  an  interposed  stratum  of 
the  unguent. 


SURFACES  OF  CONTACT. 


rallel 


Oak  upon  oak,  fibres  parall 

Ditto        ditto 

Ditto        ditto 

Ditto,  fibres  perpendicular 

Ditto        ditto 

Ditto        ditto 

Ditto  upon  elm,  fibres  parallel 

Ditto        ditto 

Ditto        ditto 

Ditto  upon  cast  iron,  ditto 

Ditto  upon  wrought  iron,  ditto 
Beech  upon  oak,  ditto 
Elm  upon  oak,  ditto  • 

Ditto        ditto 

Ditto        ditto 

Ditto  upon  elm,  ditto     • 

Ditto  upon  cast  iron,  ditto 

Wrought  iron  upon  oak,  ditto 


Ditto        ditto 

ditto  • 

Ditto        ditto 

ditto  • 

Ditto  upon  elm. 

ditto  • 

Ditto        ditto 

ditto  • 

Ditto        ditto 

ditto  • 

Ditto  upon  east  iron,  ditto 

Ditto        ditto 

ditto  • 

Ditto        ditto 

ditto  • 

Ditto  upon  wrought  iron,  ditto 

Ditto        ditto 

ditto  • 

Ditto        ditto 

ditto  • 

Wrought  iron  upon  brass,  fibres 

parallel  • 
Ditto        ditto 

ditto  • 

Ditto        ditto 

ditto  • 

Cast  iron  upon  oak 

ditto  • 

Ditto        ditto 

ditto  • 

Ditto        ditto 

ditto  • 

Ditto        ditto 

ditto  • 

Ditto        ditto 

ditto  • 

Ditto  upon  elm, 

ditto  • 

Ditto        ditto 

ditto  • 

•Ditto        ditto        ditto  • 

Ditto,  ditto  upon  wrought  iron 
Cast  iron  upon  east  iron     • 
Ditto        ditto 


Friction 

Friction 

or 

OF 

Motion. 

Quiescence. 

^ 

UNGUENTS. 

s     = 

0      = 

•r .    ® 

■  ■  ■    £ 

S':| 

(=  =  u 

S     '^ 

6   ^ 

0,164 

0,440 

Drv  soap. 

0,075 

0,164 

Tallow. 

0,067 

Ilog's  lard. 

o,o83 

0,254 

Tallow. 

0,072 

Ho^'s  lard. 

0,230 

. 

Water. 

0,1 36 

Drv  soap. 

0,073 

0,178 

Tallow. 

0,066 

Hog's  lard. 
Tallow. 

0,080 

0,098 

Tallow. 

o,o55 

Tallow. 

0,137 

0,411 

Drv  soap. 

0.070 

0,142 

Tallow. 

0,060 

Hog's  lard. 

0,139 

0,217 

Drv  soop. 

0,066 

Tallow. 

j  Greased,  and 
<  saturated  with 

0,256 

0,649 

1  water. 

0,214 

Dry  soap. 

o,o85 

0,108 

Tailow. 

0,078 

Tallow. 

0,076 

Hog's  lard. 

o,o55 

Olive  oil. 

o,io3 

Tallow. 

0.076 

Hog's  lard. 

0,066 

0,100 

Olive  oil. 

0,082 

Tallow. 

0,081 

Hoir's  lard. 

0,070 

0,1 15 

Olive  oil. 

o,io3 

Tallow. 

0,075 

Ilog's  lard. 

0,078 

Olive  oil. 

0,189 

Dry  soap. 
1  Greased,  and 

0,218 

0,646 

<  saturated  with 
1  water. 

0,078 

0,100 

Tallow. 

0,075 

ling's  lard. 

0,075 

0,100 

Olive  oil. 

0.077 

Tallow. 

0,061 

Olive  oil. 

Ilog's  lard  and 

pUimbugo. 
Tallow. 

0,091 

0,100 

o,3i4 

Water. 

0,197 

Soap. 

U2 


TABLE    VIII. 


TABLE  Yin..—co7itimied. 


SURFACES  OF  CONTACT. 


Friction 

OF 

Motion. 


5    ' 


Cast  iron  upon  cast  iron 
Ditto        ditto 
Ditto        ditto 

Ditto        ditto 

Ditto  upon  brass    • 

Ditto        ditto 

Ditto        ditto 
Copper  upon  oak,  fibres  parallel 
Yellow  copper  upon  cast  iron 

Ditto         ditto 

Ditto         ditto 
Brass  upon  cast  iron- 

Ditto        ditto 

Ditto  upon  wrought  iron 

Ditto        ditto 

Ditto        ditto 
Ditto  upon  brass 
Steel  upon  cast  iron 
Ditto        ditto 
Ditto  •      ditto 
Ditto  upon  wrought 
Ditto        ditto 
Ditto  upon  brass 
Ditto        ditto 

Ditto        ditto 

Tanned  ox  hide  upon  cast  iron 

Ditto        ditto 
Ditto        ditto 
Ditto  upon  brass 
Ditto        ditto 
Ditto  upon  oak, 

Hempen  fibres  not  twisted,  mov- 
ing upon  oak,  the  fibres  of  the 
liemp  being  placed  in  a  direc- 
tion perpendicular  to  the  direc-  f 
tion  of  the  motion,  and  those  j 
of  the  oak  parallel  to  it  •         •  ) 

The  same  as  above,  moving  upon  (^ 
cast  iron  •  •  •  •  ) 
Ditto 

Soft  calcareous  stone  of  Jaumont ") 
upon  the  same,  with  a  layer  of 
mortar,  of  sand,  and  lime  inter- 
posed, after  from  10  to  15  min- 
utes' contact. 


0,100 

0,070 
0,064 

o,o55 

o,io3 
0,075 
0,078 
0,069 
0,072 
0,068 
0.066 
0,086 
0,077 
0,081 

0,089 

0,072 
o,o58 
o,io5 
0,081 
0,079 
0,093 
0,076 
o,o56 
o,o53 

0,067 

0,365 

o,i59 
o,i33 
0,241 
0,191 
0,29 


0,332 


0,194 
0,1 53 


Friction 

OF 

Quiescence. 


0,100 
0,100 


0,100 
o,io3 


0.106 


0,79 


0,869 


UNGUENTS. 


0,74 


Tallow. 
Hogs'  lard. 
Olive  oil. 
J  Lard  and 
I  plumbago. 
Tallow. 
Hogs'  lard. 
Olive  oil. 
Tallow. 
Tallow. 
Hogs'  lard. 
Olive  oil. 
Tidlow. 
Olive  oil. 
Tallow, 
j  Lard  and 
I  plumbago. 
Olive  oil. 
Olive  oil. 
Tallow. 
Hogs'  lard. 
Olive  oil. 
Tallow. 
Hogs'  lard. 
Tallow. 
Olive  oil. 
I  Lard  and 
I  plumbago. 
Greased,  and 
saturated  with 
water. 
Tallow. 
Olive  oil. 
Tallow. 
Olive  oil. 
"Water. 

I  Greased,  and 
.<  saturated  with 
(  water. 


Tallow. 
Olive  oil. 


TABLE  IX. 


US 


TABLE    IX. 

OF  WEIGHTS  NECESSARY  TO  BEND  DIFFERENT  ROPES  AROUND  A  WHEEL 
ONE  FOOT  IN  DIAMETER. 


No.  1.   White  Ropes— new  and  dry. 
Stiffness  proportional  to  the  square  of  the  diameter. 


Diameter  of  rope 
in  inctius. 

Natural  stiffness, 
or  value  of  K. 

Stiffness  for  load  of 
1  lb.,  or  value  of  /. 

0,39 

0,79 
1,57 
3,i5 

lbs. 

0,4024 

1,6097 

6,4389 

25,7553 

lbs. 

0,0079877         1 
o,o3i95oi 

0,1278019     : 

0,5112019 

No.  2.    White  Ropes— new  and  moistened    with 

WATEK. 


Stiffness  proportional  to  square  of  diameter. 

Diameter  of  rope 
in  inches. 

Natural  stiffness, 
or  value  of  K. 

Stiffness  for  load  ol 
I  lb.,  or  value  of  /. 

0,39 
0,79 

'/I 
3,i5 

lbs. 

0,8048 

3,2194 

12,8772 

5i,5iu 

lbs. 

0,0079877 
o,o3i95oi 
0,1278019 
0,5112019 

No.  3.    White  Ropes— half  woun  and  dry. 

Stiffness  proportional  to  the  square  root  of  the  cube  of 
the  diameter. 


Diameter  of  rope 
in  inches. 

Natural  Stiffness, 
or  value  of  K. 

Stiffness  for  load  of 
1  lb.,  or  value  of  /. 

0,39 
0,79 
1,57 
3,i5 

lbs. 
0,40243 
i,i38oi 
3,21844 
9,ioi5o 

lbs. 

0,0079877 
0,0525889 
0,0638794 
0,1806573 

No.  4.  White  Ropes— half  worn  and  moistened 

WITH    water. 

stiffness  proportional  to  the  square  root  of  the  cule  of 
the  diameter. 


Diameter  of  rope 
in  inches. 

Natural  Stiffness, 
or  value  of  A'. 

Stitfness  for  lo;id  of 
1  lb.,  or  value  of  7. 

0,39 

0,79 
1,57 
3,i5 

lbs. 

0,8048 
2,2761 
6,4324 
18,2037 

lbs. 

0,0079877 

o,o525889 

0,0638794 

0,1806573 

Squares  of  the  ratios 

ol  diameter,  or  val 

ues  of  d". 

Squires 

Ratios  d. 

d\ 

1,00 

1,00 

1,10 

1,21 

1,20 

1,44 

I,3o 

1,69 

1, 40 

1,96 

i,5o 

2, 2D 

1,60 

2.56 

1,70 

2.89 

1,80 

3,24 

1,90 

3,61 

2,00 

4,00 

Square  roots  of  the 

culies  of  the  ntios 

of  diamctt-r,  or  val- 

ues of  d2' 

llatios  or 

d. 

Power  ^, 
or  d2- 

1,00 

1,000 

1,10 

1,1 54 

1,20 

i,3i5 

,  i.3o 

1,482 

1,40 

1,657 

i,5o 

1,837 

1,60 

2,024 

I.^o 

2,217 

1,80 

2,4i5 

1,90 
2,00 

2,619 
2,828 

444 


TABLE  IX. 


TABLE  IX — continued. 


'Ho.  5.  Tarred  Eopes. 

Stiff iiess  proportional  to  the  number  of  yarns. 

[These  ropes  are  usually  made  of  three  strands  twisted  around  each  other,  each  strand  being  corn- 
ed of  a  certain  number  of  yarns,  also  twisted  about  each  other  in  the  same  manner  ] 


^0.  of  yarns. 

Weight  of  1  font  in 
length  of  rope. 

Naturnl  stiffness,  or 
value   of  K. 

Stiffness   fur  hmd   of 
1  lb.,  or  Vi.lue  of  /. 

6 
l5 
3o 

lbs. 
0,0211 

0,0497 
i,oi37 

lbs. 

o,i534 
0,7664 
2,5297 

lbs. 
0,0085198 
0,0198796 
0,0411799 

TABLE   X. 


445 


TABLE    X. 

FRICTION  OF  TKUNNIONS  IN  THEIR  BOXES. 


KINDS  OF  MATERIALS. 


Ratio  of  friclion  lo 
pressure  when  the 
ungneiit  is  renewed 


STATE  OF  SURF.^CES. 


Trunnions  of  cast  iron  and 
boxes  of  cast  iron. 


Trunnions  ofcast  iron  and 
boxes  of  brass. 


Trunnions  of  cast  iron  and 
bo.xes  of  lignum-vita?. 


Trunnions  of  wrought  iron 
and  boxes  of  cast  iron. 


Truiinions  of  wrought  iron 
and  boxes  of  brass. 


Trunnions  of  wrought  iron 

and  boxes  of  liguutu-vi- 

tae. 
Trunnions   of  brass  and  ( 

boxes  of  brass.  } 

Trunnions   of  brass  and  ) 

boxes  of  cast  iron.  ) 

Trunnions  of  lignuin-vita;  < 
and  boxes  ofcast  iron.    I 

Trunnions  of  lignum-vita; 
and  boxes  of  lignum- 
vit;e. 


By  the 
orUinarj' 
method. 


Unguents  of  olive  oil,  hogs'  lard, 

and  tallow     • 
The  same  unguents  moistened  witii 

water     .         .         •         • 
Unguent  of  asphaltum 
Unctuous .         .         •         • 
Unctuous  and  moistened  with  wa- 
ter        •••         • 

Unguents  of  olive  oil,  hogs'  lard, 
and  tallow 

Unctuous  •         •   _      •         •    _     • 

Uuctuous  and  moistened  witii  wa- 
ter   

Very  slightly  unctuous 

Without  unguents     • 

Unguents  of  olive  oil  and  hogs'  ) 
lard i 

Unctuous  with  oil  and  hogs'  lard 

Unctuous  with  a  mixture  of  hogs' 
lard  and  plumbago 

Unguents  of  olive  oil,  tallow,  and 
hogs'  lard 

Unguents  of  olive  oil,  hogs'  lard, 
and  tiillow 

Old  uniruents  liardened     • 
Unctuous  and  moistened  with  wa- 
ter          

Very  slightly  unctuous 

Unguents  of  oil  or  hogs'  lard     • 
Unctuous .         .         •         •         • 

Unguent  of  oil- 
Unguent  of  hogs'  lard 

Unguents  of  tallow  or  of  olive  oil 

Unguents  of  hogs'  lard      •         •  , 
Unctuous-         .... 

Unjruent  of  hogs'  lard 


I  0,08  ) 


0,08 

0,034 

0,14 

0,14 
<  0,07    I 

i  0,08  I 
0,16 

0,16 
0,19 
0,18 


0,10 

0.14 

0,07 

to 

0,08 

to 

( 0.08  ) 

0,09 

0,10 
0,23 

0,1  I 

0,19 

0,10 
0,09 


0,12 

o,i5 


Or,  con- 
tinuously 


O,o54 


0,034 
0,034 


0,034 

0,090 

o,o54 
o,o54 


0,043  ] 

0,032  < 


0,07 


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